Not necessarily. If the income you earned in NYC was taxed by the state at only $5,000, then you would only get a credit of $5,000. If the same income would have had taxes of $8,000 in NJ, then you would still only get a credit of $5,000 and still be responsible for the other $3,000.
If this is not what is happening in your situation, you may need to go back through and make sure your income is properly allocated and that you did YOUR NEW YORK taxes first, the the credit for NY will be transferred to NJ.
Also, only the state taxes are credited, NOT the NYC taxes.
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