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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2013-10-11</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>2013-10-11</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>F</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>495</value>
                    <footnoteId id="F1"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>20.72</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>30843</value>
                    <footnoteId id="F2"/>
                    <footnoteId id="F3"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <footnotes>
        <footnote id="F1">Shares withheld to settle income tax liability on restricted stock that vested on October 11, 2013.</footnote>
        <footnote id="F2">Total includes 4,975 shares of unvested restricted stock granted 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="F3">Total includes 1,250 shares of unvested restricted stock granted under the National Bank Holdings Corporation 2009 Equity Incentive Plan that are scheduled to vest on October 11, 2014.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Zsolt K. Bessko as Attorney-in-Fact</signatureName>
        <signatureDate>2013-10-21</signatureDate>
    </ownerSignature>
</ownershipDocument>
