Yes, if you file a New Jersey resident tax return and you pay state tax to a different state for the same income, (not the amount withheld, the amount of the tax liability) you will see a credit for that double taxation on your New Jersey return.
To have the numbers flow properly in TurboTax, the non-resident return must be done first, then the tax on the income claimed a second time on the New Jersey return will be credited, up to the same amount New Jersey would assess.
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