It depends. You certainly will want to get the NY W-4 on file with your Jersey employer to prevent tax shortfalls. Here is what happens on your return:
You are taxed by both states: NY and NJ. NY (and NYC) tax you for being a resident, while NJ taxes you as a nonresident. You prepare your NJ return first, as NY will give you credit for the amount of tax you must pay to NJ on the same income.
From what I'm familiar with, NYS does have a slightly higher tax rate than NJ, and as a NYC resident you have an additional tax for NYC. Therefore, if no NY tax was taken out of your check, it is possible that you do have an amount to owe, which could be 4,000, but first you want to see if the NY return is giving credit for NJ taxes. Your NJ taxes are the amount of NJ tax withheld on your W-2, minus any refund or including any additional amount due.
Before accepting the amount, therefore, I recommend deleting the NY return temporarily and making sure the NJ return is complete first. Then add the NY return and make sure as you go through the screens that the credit for NJ is calculated. If you've done that and your amount due is still 4,000, then it probably is the correct amount. (I'm willing to venture, however, that you will see the amount due to NY go down).
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