<ownershipDocument>
    <schemaVersion>X0201</schemaVersion>

    <documentType>4</documentType>
<periodOfReport>1998-01-01</periodOfReport>
<notSubjectToSection16>0</notSubjectToSection16>
    <issuer>
        <issuerCik>0000011199</issuerCik>
        <issuerName>BEMIS CO INC</issuerName>
        <issuerTradingSymbol>BMS</issuerTradingSymbol>
</issuer>
<reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001215482</rptOwnerCik>
            <rptOwnerName>THEISEN HENRY J</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>SUITE 2300</rptOwnerStreet1>
            <rptOwnerStreet2>222 S. 9TH ST.</rptOwnerStreet2>
            <rptOwnerCity>MINNEAPOLIS</rptOwnerCity>
            <rptOwnerState>MN</rptOwnerState>
            <rptOwnerZipCode>55402-4099</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
            <reportingOwnerRelationship>
<isDirector>0</isDirector>
<isOfficer>1</isOfficer>
<isTenPercentOwner>0</isTenPercentOwner>
<isOther>0</isOther>
<officerTitle>Executive Vice President</officerTitle>
<otherText></otherText>
</reportingOwnerRelationship>
</reportingOwner>
<nonDerivativeTable><nonDerivativeTransaction><securityTitle><value>Common Stock</value>
</securityTitle>
<transactionDate><value>2004-01-02</value>
</transactionDate>
<transactionCoding><transactionFormType>4</transactionFormType>
<transactionCode>M</transactionCode>
<equitySwapInvolved>0</equitySwapInvolved>
</transactionCoding>
<transactionAmounts><transactionShares><value>7906</value>
<footnoteId id="F4"></footnoteId>
</transactionShares>
<transactionPricePerShare><value>49.51</value>
</transactionPricePerShare>
<transactionAcquiredDisposedCode><value>A</value>
</transactionAcquiredDisposedCode>
</transactionAmounts>
<postTransactionAmounts><sharesOwnedFollowingTransaction><value>7906</value>
</sharesOwnedFollowingTransaction>
</postTransactionAmounts>
<ownershipNature><directOrIndirectOwnership><value>D</value>
</directOrIndirectOwnership>
</ownershipNature>
</nonDerivativeTransaction>
<nonDerivativeTransaction><securityTitle><value>Common Stock</value>
</securityTitle>
<transactionDate><value>2003-03-31</value>
</transactionDate>
<transactionCoding><transactionFormType>4</transactionFormType>
<transactionCode>J</transactionCode>
<equitySwapInvolved>0</equitySwapInvolved>
</transactionCoding>
<transactionAmounts><transactionShares><value>103</value>
</transactionShares>
<transactionPricePerShare><footnoteId id="F1"></footnoteId>
</transactionPricePerShare>
<transactionAcquiredDisposedCode><value>A</value>
</transactionAcquiredDisposedCode>
</transactionAmounts>
<postTransactionAmounts><sharesOwnedFollowingTransaction><value>4900</value>
</sharesOwnedFollowingTransaction>
</postTransactionAmounts>
<ownershipNature><directOrIndirectOwnership><value>I</value>
</directOrIndirectOwnership>
<natureOfOwnership><value>401(k) Plan</value>
</natureOfOwnership>
</ownershipNature>
</nonDerivativeTransaction>
<nonDerivativeTransaction><securityTitle><value>Common Stock</value>
</securityTitle>
<transactionDate><value>2003-06-30</value>
</transactionDate>
<transactionCoding><transactionFormType>4</transactionFormType>
<transactionCode>J</transactionCode>
<equitySwapInvolved>0</equitySwapInvolved>
</transactionCoding>
<transactionAmounts><transactionShares><value>48</value>
</transactionShares>
<transactionPricePerShare><footnoteId id="F1"></footnoteId>
</transactionPricePerShare>
<transactionAcquiredDisposedCode><value>A</value>
</transactionAcquiredDisposedCode>
</transactionAmounts>
<postTransactionAmounts><sharesOwnedFollowingTransaction><value>4948</value>
</sharesOwnedFollowingTransaction>
</postTransactionAmounts>
<ownershipNature><directOrIndirectOwnership><value>I</value>
</directOrIndirectOwnership>
<natureOfOwnership><value>401(k) Plan</value>
</natureOfOwnership>
</ownershipNature>
</nonDerivativeTransaction>
<nonDerivativeTransaction><securityTitle><value>Common Stock</value>
</securityTitle>
<transactionDate><value>2003-09-30</value>
</transactionDate>
<transactionCoding><transactionFormType>4</transactionFormType>
<transactionCode>J</transactionCode>
<equitySwapInvolved>0</equitySwapInvolved>
</transactionCoding>
<transactionAmounts><transactionShares><value>72</value>
</transactionShares>
<transactionPricePerShare><footnoteId id="F1"></footnoteId>
</transactionPricePerShare>
<transactionAcquiredDisposedCode><value>A</value>
</transactionAcquiredDisposedCode>
</transactionAmounts>
<postTransactionAmounts><sharesOwnedFollowingTransaction><value>5020</value>
</sharesOwnedFollowingTransaction>
</postTransactionAmounts>
<ownershipNature><directOrIndirectOwnership><value>I</value>
</directOrIndirectOwnership>
<natureOfOwnership><value>401(k) Plan</value>
</natureOfOwnership>
</ownershipNature>
</nonDerivativeTransaction>
<nonDerivativeTransaction><securityTitle><value>Common Stock</value>
</securityTitle>
<transactionDate><value>2003-12-31</value>
</transactionDate>
<transactionCoding><transactionFormType>4</transactionFormType>
<transactionCode>J</transactionCode>
<equitySwapInvolved>0</equitySwapInvolved>
</transactionCoding>
<transactionAmounts><transactionShares><value>27</value>
</transactionShares>
<transactionPricePerShare><footnoteId id="F1"></footnoteId>
</transactionPricePerShare>
<transactionAcquiredDisposedCode><value>A</value>
</transactionAcquiredDisposedCode>
</transactionAmounts>
<postTransactionAmounts><sharesOwnedFollowingTransaction><value>5047</value>
</sharesOwnedFollowingTransaction>
</postTransactionAmounts>
<ownershipNature><directOrIndirectOwnership><value>I</value>
</directOrIndirectOwnership>
<natureOfOwnership><value>401(k) Plan</value>
</natureOfOwnership>
</ownershipNature>
</nonDerivativeTransaction>
</nonDerivativeTable>
<derivativeTable><derivativeTransaction><securityTitle><value>Common Stock</value>
</securityTitle>
<conversionOrExercisePrice><footnoteId id="F2"></footnoteId>
</conversionOrExercisePrice>
<transactionDate><value>1998-01-01</value>
</transactionDate>
<deemedExecutionDate><value>2004-01-02</value>
</deemedExecutionDate>
<transactionCoding><transactionFormType>4</transactionFormType>
<transactionCode>M</transactionCode>
<equitySwapInvolved>0</equitySwapInvolved>
</transactionCoding>
<transactionAmounts><transactionShares><value>12000</value>
</transactionShares>
<transactionPricePerShare><value>49.51</value>
</transactionPricePerShare>
<transactionAcquiredDisposedCode><value>D</value>
</transactionAcquiredDisposedCode>
</transactionAmounts>
<exerciseDate><value>2003-12-31</value>
<footnoteId id="F4"></footnoteId>
</exerciseDate>
<expirationDate><value>2003-12-31</value>
</expirationDate>
<underlyingSecurity><underlyingSecurityTitle><value>Common Stock</value>
</underlyingSecurityTitle>
<underlyingSecurityShares><value>12000</value>
</underlyingSecurityShares>
</underlyingSecurity>
<postTransactionAmounts><sharesOwnedFollowingTransaction><value>0</value>
</sharesOwnedFollowingTransaction>
</postTransactionAmounts>
<ownershipNature><directOrIndirectOwnership><value>D</value>
</directOrIndirectOwnership>
</ownershipNature>
</derivativeTransaction>
<derivativeTransaction><securityTitle><value>Common Stock</value>
</securityTitle>
<conversionOrExercisePrice><footnoteId id="F2"></footnoteId>
</conversionOrExercisePrice>
<transactionDate><value>2001-01-01</value>
</transactionDate>
<transactionCoding><transactionFormType>4</transactionFormType>
<transactionCode>A</transactionCode>
<equitySwapInvolved>0</equitySwapInvolved>
</transactionCoding>
<transactionAmounts><transactionShares><value>25000</value>
</transactionShares>
<transactionPricePerShare><footnoteId id="F3"></footnoteId>
</transactionPricePerShare>
<transactionAcquiredDisposedCode><value>A</value>
</transactionAcquiredDisposedCode>
</transactionAmounts>
<exerciseDate><value>2005-12-31</value>
<footnoteId id="F5"></footnoteId>
</exerciseDate>
<expirationDate><value>2005-12-31</value>
</expirationDate>
<underlyingSecurity><underlyingSecurityTitle><value>Common Stock</value>
</underlyingSecurityTitle>
<underlyingSecurityShares><value>25000</value>
</underlyingSecurityShares>
</underlyingSecurity>
<postTransactionAmounts><sharesOwnedFollowingTransaction><value>25000</value>
</sharesOwnedFollowingTransaction>
</postTransactionAmounts>
<ownershipNature><directOrIndirectOwnership><value>D</value>
</directOrIndirectOwnership>
</ownershipNature>
</derivativeTransaction>
<derivativeTransaction><securityTitle><value>Common Stock</value>
</securityTitle>
<conversionOrExercisePrice><footnoteId id="F2"></footnoteId>
</conversionOrExercisePrice>
<transactionDate><value>2002-01-01</value>
</transactionDate>
<transactionCoding><transactionFormType>4</transactionFormType>
<transactionCode>A</transactionCode>
<equitySwapInvolved>0</equitySwapInvolved>
</transactionCoding>
<transactionAmounts><transactionShares><value>8497</value>
</transactionShares>
<transactionPricePerShare><footnoteId id="F3"></footnoteId>
</transactionPricePerShare>
<transactionAcquiredDisposedCode><value>A</value>
</transactionAcquiredDisposedCode>
</transactionAmounts>
<exerciseDate><value>2006-12-31</value>
<footnoteId id="F6"></footnoteId>
</exerciseDate>
<expirationDate><value>2006-12-31</value>
</expirationDate>
<underlyingSecurity><underlyingSecurityTitle><value>Common Stock</value>
</underlyingSecurityTitle>
<underlyingSecurityShares><value>8497</value>
</underlyingSecurityShares>
</underlyingSecurity>
<postTransactionAmounts><sharesOwnedFollowingTransaction><value>8497</value>
</sharesOwnedFollowingTransaction>
</postTransactionAmounts>
<ownershipNature><directOrIndirectOwnership><value>D</value>
</directOrIndirectOwnership>
</ownershipNature>
</derivativeTransaction>
<derivativeTransaction><securityTitle><value>Common Stock</value>
</securityTitle>
<conversionOrExercisePrice><footnoteId id="F2"></footnoteId>
</conversionOrExercisePrice>
<transactionDate><value>2003-01-02</value>
</transactionDate>
<transactionCoding><transactionFormType>4</transactionFormType>
<transactionCode>A</transactionCode>
<equitySwapInvolved>0</equitySwapInvolved>
</transactionCoding>
<transactionAmounts><transactionShares><value>12012</value>
</transactionShares>
<transactionPricePerShare><footnoteId id="F3"></footnoteId>
</transactionPricePerShare>
<transactionAcquiredDisposedCode><value>A</value>
</transactionAcquiredDisposedCode>
</transactionAmounts>
<exerciseDate><value>2007-12-31</value>
<footnoteId id="F7"></footnoteId>
</exerciseDate>
<expirationDate><value>2007-12-31</value>
</expirationDate>
<underlyingSecurity><underlyingSecurityTitle><value>Common Stock</value>
</underlyingSecurityTitle>
<underlyingSecurityShares><value>12012</value>
</underlyingSecurityShares>
</underlyingSecurity>
<postTransactionAmounts><sharesOwnedFollowingTransaction><value>12012</value>
</sharesOwnedFollowingTransaction>
</postTransactionAmounts>
<ownershipNature><directOrIndirectOwnership><value>D</value>
</directOrIndirectOwnership>
</ownershipNature>
</derivativeTransaction>
<derivativeTransaction><securityTitle><value>Common Stock</value>
</securityTitle>
<conversionOrExercisePrice><value>49.18</value>
</conversionOrExercisePrice>
<transactionDate><value>2002-01-01</value>
</transactionDate>
<transactionCoding><transactionFormType>4</transactionFormType>
<transactionCode>A</transactionCode>
<equitySwapInvolved>0</equitySwapInvolved>
</transactionCoding>
<transactionAmounts><transactionShares><value>11179</value>
</transactionShares>
<transactionPricePerShare><footnoteId id="F9"></footnoteId>
</transactionPricePerShare>
<transactionAcquiredDisposedCode><value>A</value>
</transactionAcquiredDisposedCode>
</transactionAmounts>
<exerciseDate><footnoteId id="F8"></footnoteId>
</exerciseDate>
<expirationDate><value>2011-12-31</value>
</expirationDate>
<underlyingSecurity><underlyingSecurityTitle><value>Common Stock</value>
</underlyingSecurityTitle>
<underlyingSecurityShares><value>11179</value>
</underlyingSecurityShares>
</underlyingSecurity>
<postTransactionAmounts><sharesOwnedFollowingTransaction><value>11179</value>
</sharesOwnedFollowingTransaction>
</postTransactionAmounts>
<ownershipNature><directOrIndirectOwnership><value>D</value>
</directOrIndirectOwnership>
</ownershipNature>
</derivativeTransaction>
<derivativeTransaction><securityTitle><value>Common Stock</value>
</securityTitle>
<conversionOrExercisePrice><value>49.63</value>
</conversionOrExercisePrice>
<transactionDate><value>2003-01-02</value>
</transactionDate>
<transactionCoding><transactionFormType>4</transactionFormType>
<transactionCode>A</transactionCode>
<equitySwapInvolved>0</equitySwapInvolved>
</transactionCoding>
<transactionAmounts><transactionShares><value>15804</value>
</transactionShares>
<transactionPricePerShare><footnoteId id="F9"></footnoteId>
</transactionPricePerShare>
<transactionAcquiredDisposedCode><value>A</value>
</transactionAcquiredDisposedCode>
</transactionAmounts>
<exerciseDate><footnoteId id="F8"></footnoteId>
</exerciseDate>
<expirationDate><value>2012-12-31</value>
</expirationDate>
<underlyingSecurity><underlyingSecurityTitle><value>Common Stock</value>
</underlyingSecurityTitle>
<underlyingSecurityShares><value>15804</value>
</underlyingSecurityShares>
</underlyingSecurity>
<postTransactionAmounts><sharesOwnedFollowingTransaction><value>15804</value>
</sharesOwnedFollowingTransaction>
</postTransactionAmounts>
<ownershipNature><directOrIndirectOwnership><value>D</value>
</directOrIndirectOwnership>
</ownershipNature>
</derivativeTransaction>
</derivativeTable>
<footnotes><footnote id="F1">No price necessary for Edgar Filing - Company 401(k) Plan.</footnote>
<footnote id="F2">Security converts to Common Stock on a one-for-one basis on date of conversion.</footnote>
<footnote id="F3">Will know price on the date of conversion.</footnote>
<footnote id="F4">Award under Bemis Stock Award Program exempt under Rule 16b-3:  Grant to Reporting Person of right to receive stock as of December 31, 2003.  On January 2, 2004 payout was made, with 4,094 shares withheld for tax purposes, leaving right to receive 7,906 shares.</footnote>
<footnote id="F5">Award under Bemis Time Accelerated Restricted Stock Appreciation Program (TARSAP) exempt under Rule 16b-3: Grant to Reporting Person of right to receive stock as of December 31, 2005, or if certain performance targets are met by the Company on December 31, 2003.</footnote>
<footnote id="F6">Award under Bemis Time Accelerated Restricted Stock Appreciation Program (TARSAP) exempt under Rule 16b-3: Grant to Reporting Person of right to receive stock as of December 31, 2006, or if certain performance targets are met by the Company on December 31, 2004.</footnote>
<footnote id="F7">Award under Bemis Time Accelerated Restricted Stock Appreciation Program (TARSAP) exempt under Rule 16b-3: Grant to Reporting Person of right to receive stock as of December 31, 2007, or if certain performance targets are met by the Company on December 31, 2005.</footnote>
<footnote id="F8">Stock Option (right to buy) pursuant to Stock Option Plan exempt under Rule 16b-3: Options exercisable in three increments of 33 percent on the first, second and third anniversary of date of grant.</footnote>
<footnote id="F9">Will know price on the date of exercise.</footnote>
</footnotes>
<ownerSignature><signatureName>J J Seifert Power of Attorney</signatureName>
<signatureDate>2004-01-06</signatureDate>
</ownerSignature>
</ownershipDocument>

