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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2017-05-25</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001444380</issuerCik>
        <issuerName>NEVRO CORP</issuerName>
        <issuerTradingSymbol>NVRO</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001190474</rptOwnerCik>
            <rptOwnerName>JAEGER WILFRED E</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O THREE ARCH PARTNERS</rptOwnerStreet1>
            <rptOwnerStreet2>19 SOUTH B STREET, SUITE 14</rptOwnerStreet2>
            <rptOwnerCity>SAN MATEO</rptOwnerCity>
            <rptOwnerState>CA</rptOwnerState>
            <rptOwnerZipCode>94401</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle></officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2017-05-25</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>1518</value>
                    <footnoteId id="F1"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0.00</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>1518</value>
                    <footnoteId id="F2"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>1125291</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>See footnote</value>
                    <footnoteId id="F3"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>24847</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>See footnote</value>
                    <footnoteId id="F4"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
    </nonDerivativeTable>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Stock Option (Right to Buy)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>74.07</value>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2017-05-25</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>3545</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0.00</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <footnoteId id="F5"/>
            </exerciseDate>
            <expirationDate>
                <value>2027-05-24</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>3545</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>3545</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Constitute restricted stock units (&quot;RSUs&quot;) for which the Reporting Person is entitled to receive one (1) share of Common Stock for each RSU upon vesting. 100% of the RSUs shall vest on the one year anniversary of May 25, 2017, subject to the Reporting Person's continued service to the issuer.</footnote>
        <footnote id="F2">Includes 1,518 RSUs.</footnote>
        <footnote id="F3">Shares are held by Three Arch Partners IV, L.P. (&quot;TAP IV&quot;). Three Arch Management IV, L.L.C. (&quot;TAM IV&quot;), the general partner of TAP IV, has sole voting and dispositive power with respect to the securities held by TAP IV. Mark A. Wan and Wilfred E. Jaeger, the managing members of TAM IV, share the voting and dispositive power with respect to the shares held by TAP IV. Such persons and entities disclaim beneficial ownership of the shares held by TAP IV except to the extent of any pecuniary interest therein.</footnote>
        <footnote id="F4">Shares are held by Three Arch Associates IV, L.P. (&quot;TAA IV&quot;). TAM IV, the general partner of TAA IV, has sole voting and dispositive power with respect to the securities held by TAA IV. Mark A. Wan and Wilfred E. Jaeger, the managing members of TAM IV, share the voting and dispositive power with respect to the shares held by TAA IV. Such persons and entities disclaim beneficial ownership of the shares held by TAA IV except to the extent of any pecuniary interest therein.</footnote>
        <footnote id="F5">1/12th of the shares subject to the option vest on each monthly anniversary measured from May 25, 2017 (the &quot;Vesting Commencement Date&quot;), such that 100% of the shares subject to the option will be fully vested and exercisable on the first anniversary of the Vesting Commencement Date, subject to the Reporting Person's continued service to the issuer.</footnote>
    </footnotes>

    <ownerSignature>
        <signatureName>/s/ Rami Elghandour, as Attorney-in-Fact for Wilfred E. Jaeger</signatureName>
        <signatureDate>2017-05-30</signatureDate>
    </ownerSignature>
</ownershipDocument>
