<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0304</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2011-12-30</periodOfReport>

    <notSubjectToSection16>1</notSubjectToSection16>

    <issuer>
        <issuerCik>0001109525</issuerCik>
        <issuerName>BANCORP RHODE ISLAND INC</issuerName>
        <issuerTradingSymbol>BARI</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001444340</rptOwnerCik>
            <rptOwnerName>Sy Tiffany R</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>122 PILGRIM DRIVE</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>WARWICK</rptOwnerCity>
            <rptOwnerState>RI</rptOwnerState>
            <rptOwnerZipCode>02888</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Controller</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2012-01-01</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>D</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>100</value>
                </transactionShares>
                <transactionPricePerShare>
                    <footnoteId id="F1"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>0</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Employee Stock Option (right to buy)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>19.00</value>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2011-12-30</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>D</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>150</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>29.25</value>
                    <footnoteId id="F2"/>
                    <footnoteId id="F3"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <value>2010-04-21</value>
            </exerciseDate>
            <expirationDate>
                <value>2012-01-01</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>150</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>0</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Disposed of pursuant to the Agreement and Plan of Merger, dated as of April 19, 2011 (the &quot;Merger Agreement&quot;), between Bancorp Rhode Island, Inc. (&quot;BancorpRI&quot;) and Brookline Bancorp, Inc. (&quot;Brookline&quot;). Pursuant to the Merger Agreement, in exchange for such BancorpRI common stock the reporting person received 4.686 shares of Brookline common stock rounded down to the nearest whole share, having a market value of $8.44 per share on 12/30/2011, for approximately 47.6% of their BancorpRI common stock and cash in the amount of $48.25 per share for the remaining 52.4% of the reporting person's BancorpRI common stock, plus cash for any fractional share based on the average closing price of Brookline common stock for the ten consecutive days ending on December 23, 2011, for total consideration of $4,410.85.</footnote>
        <footnote id="F2">The option was cancelled in connection with the merger of BancorpRI with and into Brookline pursuant to the terms of the Merger Agreement.</footnote>
        <footnote id="F3">The reporting person received $4387.50 as consideration for the cancellation.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>Margaret D. Farrell (Attorney-in-fact for Tiffany R. Sy)</signatureName>
        <signatureDate>2012-01-04</signatureDate>
    </ownerSignature>
</ownershipDocument>
