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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2020-03-27</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000737468</issuerCik>
        <issuerName>WASHINGTON TRUST BANCORP INC</issuerName>
        <issuerTradingSymbol>WASH</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001577608</rptOwnerCik>
            <rptOwnerName>Gormley Debra A.</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>1949 FLAT RIVER ROAD</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>COVENTRY</rptOwnerCity>
            <rptOwnerState>RI</rptOwnerState>
            <rptOwnerZipCode>02816</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>EVP &amp; Chief Retail Banking Off</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2020-03-27</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>2346</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>7659.723</value>
                    <footnoteId id="F1"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2020-03-27</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>F</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>1048</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>35.28</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>6611.723</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable>
        <derivativeHolding>
            <securityTitle>
                <value>Stock Options (Right to Buy)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>21.71</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <value>2014-06-13</value>
            </exerciseDate>
            <expirationDate>
                <value>2021-06-13</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>2500.0</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2500</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>Stock Options (Right to Buy)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>23.27</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <value>2015-06-18</value>
            </exerciseDate>
            <expirationDate>
                <value>2022-06-18</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>4000.0</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>4000</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>Stock Options (Right to Buy)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>32.77</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <value>2016-10-15</value>
            </exerciseDate>
            <expirationDate>
                <value>2023-10-15</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>2200.0</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2200</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>Stock Options (Right to Buy)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>32.74</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <value>2017-10-09</value>
            </exerciseDate>
            <expirationDate>
                <value>2024-10-09</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>950.0</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>950</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Balance reflects acquisitions pursuant to dividend reinvestments exempt from Form 4 reporting under Rule 16a-11.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Kristen L. DiSanto, Attorney-in-Fact</signatureName>
        <signatureDate>2020-03-31</signatureDate>
    </ownerSignature>
</ownershipDocument>
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X0306
4
2020-03-27
0
0000737468
WASHINGTON TRUST BANCORP INC
WASH
0001577608
Gormley Debra A.
1949 FLAT RIVER ROAD
COVENTRY
RI
02816
0
1
0
0
EVP & Chief Retail Banking Off
Common Stock
2020-03-27
4
A
0
2346
0
A
7659.723
D
Common Stock
2020-03-27
4
F
0
1048
35.28
D
6611.723
D
Stock Options (Right to Buy)
21.71
2014-06-13
2021-06-13
Common Stock
2500.0
2500
D
Stock Options (Right to Buy)
23.27
2015-06-18
2022-06-18
Common Stock
4000.0
4000
D
Stock Options (Right to Buy)
32.77
2016-10-15
2023-10-15
Common Stock
2200.0
2200
D
Stock Options (Right to Buy)
32.74
2017-10-09
2024-10-09
Common Stock
950.0
950
D
Balance reflects acquisitions pursuant to dividend reinvestments exempt from Form 4 reporting under Rule 16a-11.
/s/ Kristen L. DiSanto, Attorney-in-Fact
2020-03-31