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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2013-05-02</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001475841</issuerCik>
        <issuerName>National Bank Holdings Corp</issuerName>
        <issuerTradingSymbol>NBHC</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001558071</rptOwnerCik>
            <rptOwnerName>Spring Maria F</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>7800 ORCHARD RD., SUITE 300</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>GREENWOOD VILLAGE</rptOwnerCity>
            <rptOwnerState>CO</rptOwnerState>
            <rptOwnerZipCode>80111</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2013-05-02</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>4975</value>
                    <footnoteId id="F1"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0.00</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>31338</value>
                    <footnoteId id="F2"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <footnotes>
        <footnote id="F1">Shares acquired pursuant to a grant of restricted stock under the National Bank Holdings Corporation 2009 Equity Incentive Plan. Half of the shares vest on October 29, 2013 and half vest on the day before National Bank Holdings Corporation's 2014 Annual Meeting of Shareholders.</footnote>
        <footnote id="F2">Total includes 7,475 shares of unvested restricted stock granted under the National Bank Holdings Corporation 2009 Equity Incentive Plan, 4,975 of which are scheduled to vest as indicated in Footnote 1, 1,250 of which are scheduled to vest on October 11, 2013 and 1,250 of which are scheduled to vest on October 11, 2014.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/S/ Mark W. Yonkman, as Attorney-in-Fact</signatureName>
        <signatureDate>2013-05-06</signatureDate>
    </ownerSignature>
</ownershipDocument>
