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The short answer is that you always owe child support regardless of whether a case is open, whether you have visitation or parental rights, or whether or not the mother/custodial parent wants the money. The only way that you're ever relieved of your obligation to pay child support is if the child is adopted.
However, the fact that you haven't been able to pay because you didn't know where they were, because the case was closed, etc. is helpful to you as it shows that it's not really your fault that payments haven't been made.
But, you do need to try to work something out to arrange payments, etc.
This is not true under Texas law. In Texas, if any back support payments (arrears) are owed, the court retains jurisdiction to take enforcement action until the arrears are paid in full - there is no time frame that cuts off back support and there's no deadline to enforce and collect child support. http://www.supportcollectors.com/resources_texas.php
You can look at the laws for each state here: http://www.child-support-collections.com/statute-of-limitations.html
Let me ask this. If I own a company and have my current position set up on a corp to corp basis, they can only garnish from the amount the company pays me correct?