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

    <schemaVersion>X0303</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2011-01-12</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001328650</issuerCik>
        <issuerName>Bronco Drilling Company, Inc.</issuerName>
        <issuerTradingSymbol>BRNC</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001335962</rptOwnerCik>
            <rptOwnerName>Graves Zachary M</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O BRONCO DRILLING COMPANY, INC.</rptOwnerStreet1>
            <rptOwnerStreet2>16217 NORTH MAY AVENUE</rptOwnerStreet2>
            <rptOwnerCity>EDMOND</rptOwnerCity>
            <rptOwnerState>OK</rptOwnerState>
            <rptOwnerZipCode>73013</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Executive VP of Operations</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2011-01-12</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>102700</value>
                </transactionShares>
                <transactionPricePerShare>
                    <footnoteId id="F1"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>521943</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">On January 12, 2011, the reporting person received 102,700 shares of restricted stock granted under the Bronco Drilling Company, Inc. 2006 Stock Incentive Plan, as amended (the &quot;Plan&quot;).  Of these shares, 1/3 vest on February 25, 2012, 1/3 vest on February 25, 2013, and the remaining 1/3 vest on February 25, 2014; however, all unvested shares will vest immediately prior to a Change of Control (as defined in the Plan).</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>Zachary M. Graves</signatureName>
        <signatureDate>2011-01-14</signatureDate>
    </ownerSignature>
</ownershipDocument>
