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I am sorry to hear about this situation. On this website, I do not always get to give good news, and this is one of these times. First of all, courts have serious issues finding such contracts binding. This is because the Court is the proper authority to decide what is "in the best interest of the child," and thus, any such agreement has to be ratified by the court generally to be binding. See MTR OF SHAN F v. FRANCIS F, 88 Misc. 2d 165 - NY: Family Court 1976 for general discussion.
So first, someone in your situation would have to jump that hurdle. It is very hard, and the Court normally tries to void the contract. For example, did you have consideration in your contract? If not, then the contract is void.
Second, even if it is not void, then the court may look at this as a civil contract. If so, it is only enforceable for six years under N.Y. C.P.L.R. § 213(2). So you'd only be able to collect for the last 6 years. In some cases (very few), such contracts have been recognized by the courts as legitimate child support agreements, in which case the statute of limitations reaches back 20 years for anything after 1987. However the problem remains that (1) this may not be valid, and, (2) likely if it is, it can only reach back 6 years.
The courts look at this on a case by case basis. NY does not have a strong precedent for recognizing such contracts and the common law doctrine is NOT to recognize them, so it would be an uphill battle every step of the way.
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