Do Child Support Arrears Have A Statute Of Limitations In NYS?

Contact your local child support office.  Unless a court order is modified it stands.  Sounds like the state is coming after you, not something the ex initiated.  If she received state benefits and an order of support was issued or pending and not receiving money, the state will come after who the support order was issued to (namely you).  What a lot of people don't realize is that you are under an obligation to support your children even if you haven't been to court yet.  If she was receiving state assistance, they are going to look to you for at least partial repayment, since an order of support was issued.  If you gave her money for child support pending your court date, I hope you have receipts or cancelled checks.  Otherwise, you are probably stuck.  That's the way it works where I live anyway, I know it varies state to state. 

It is my understanding that there is not statute of limitation on child support arrearages.
I recently heard of a man that was injured (permanently) and awarded early Social Security benefits. In his case he had not paid support for a number of years to his ex-wife, and even though his children were in their 30s, when he received his first SSA check it was half of his calculated benefit with an explanation that since he had the arrearage which was still on the books after so many years, SSA has awarded his ex-wife 1/2 of his benefits for the rest of her life.
I say check with every agency, get a balance in writing, get it paid in full, then get a receipt reflecting a zero balance, then see if you can get a court document stating that all obligations have been fulfilled and the court order for support is no longer binding.