What is the statute of limitations on contesting the paternity of a
child born within a marriage in New York State.  In
other words, if a child is born within a marriage and subsequently I
find out that I am not the biological father, how many years can go by
before I
legally disclaim responsibility for this child?

Hello xl12-ga,

My preliminary research shows that New York laws regarding presumption
of legitimacy for a child born in wedlock are rather complex. I hope
someone can answer you question. In the meantime, here is an
interesting article to help you with understanding all the legal
issues involved in your question.

Good luck.

~ czh ~

http://www.ancpr.org/why_truth_is_not_a_defense_in_pa.htm
Why truth is not a defense in paternity actions 

Under New York law, the presumption of legitimacy can be rebutted with
DNA tests that conclusively exclude the former husband as the father
of the child.53 Conversely, New York courts also recognize paternity
by estoppel, which excludes scientific evidence of non-paternity.54
However, under New York law, neither the presumption, nor the DNA
tests, or the estoppel doctrine is regarded as absolute. Instead, each
evidentiary value is factored into a determination that is intended to
meet the best interests of the child.55 Consequently, New York courts
will admit or exclude DNA tests and will apply or not apply the
presumption of legitimacy or the estoppel doctrine based on whether
such information will assist the court in arriving at a resolution
that serves the best interests of the child.56 Hence, if a substantial
parent-child relationship has developed between the husband and the
child and no biological father is available to tag with the costs of
the child's upbringing, the New York courts may find that it is not in
the child's best interests to admit DNA evidence that disproves the
husband's paternity.57 Similarly, even if the biological father is
available, New York courts may exclude DNA evidence that proves the
biological father's paternity or that disproves the husband's
paternity on the ground that forcing a father-child relationship on
the unwilling parties would be detrimental to the child.58

The article sums it up quite correctly.  If you allowed your name on
the birth certificate and you've been in the house while the child has
been raised, it will depend entirely on the judge.  Get a good lawyer
is probably the best advice.  If you are upset enough, you can sue the
mother for the deception.   But that would be frowned upon (depending
on the age).

Youth/family courts operate very differently from normal courts.  The
judge has a lot of lee way, and always is looking for what's best. 
Hopefully, you can find the other man and make a demand that you're
not responsible financially and that the burden is on him (if he's
able to give the same level of support as you).  However, if the
mother makes enough money, then you could get off, but that's unlikely
if you cannot find the other man.

I don't know what kind of answer you're expecting, but a family lawyer
will be able to go through your options a lot more completely.

The statute of limitations to determine Paternity in NY is 21 years
(Article 5 of the Family Court Act)

A good resource (but do look around as I cannot vouch for accuracy):
http://www.divorcesource.com/NY/info/resources.shtml