In California, generally no (there are situations—such as adoption by estoppel where a non-bio dad may be stuck with a Child Support Order, but that is for a different post). However, once there has been a Judgment of Paternity, even having a DNA test that clearly “opps you out” as the child’s father is not a perennial get out jail card that you can use at any time.
California Family Code §7645 provides very specific time periods for when a motion to set aside a Judgment for Paternity has to be made. Remember a very simple rule–if you are served with a Complaint to Establish Parentage and for Child Support from DCSS— DO NOT BLOW IT OFF!!! That is the time to get some legal representation so that the situation can be handled and that you don’t wind up with an arrears order for Child Support that is impossible to ever pay down, and the child may not even be yours!!!
The recent case of San Mateo County Department of Child Support Services v. Clark (2008) 168 Cal.App.4th 834, 85 Cal.Rptr.3d 763 discussed things you can do in furtherance of a motion pursuant to Family Code §7645 ( i.e. using a deposition subpoena to compel genetic testing) …but its preferable not to have to even go there by addressing the matter before it ever gets to a Judgment.