Official NJ guidance here: https://www.nj.gov/treasury/taxation/njit14.shtml

There are a couple of pubs at the bottom with detailed information about how the credits are calculated for wage and non-wage income.

Not 100% sure about NJ, but most states only allow taxes paid to other states for residents of their state. Non residents, and part time residents do not qualify

Do you have any idea which other states you are trying to credit?

Maybe it's the other state that should be giving credit for NJ tax.

What is your client's state of residence?

What is the state whose credit you are trying to claim?  

Did they actually file returns in those either states?  Most of the time you actually need to file. Merely having withholding for another state doesn't count.

The NJ notice should explain why the credit was disallowed.

Most of the time, if we have an other state tax credit denied, it is because the state requires a copy of the other states return and we did not include it.  Lacerte tries to tell us when to attach, but sometimes it misses giving us a diagnostic telling us to do so.  Confirm you included the other state and talk to the state who denied and ask them.  They should tell you.  

As it turns out, the State reversed their decision after a review of the facts.At least that's what I've been told.

Good news. But why they disallowed the credit in the first place is a mystery. (The original auditor has retired and didn't leave an explanatory note.)

FYI, both NJ and PA often "cannot find" tax copies from the other state, even when we attach PDFs to the home state e-file. Very convenient for them.

I used to need to mail in paper copies.