Can a parent that is behind in child support get joint custody of a child whos mother already has full legal custody?

Certainly some evidence of responsibility (ie paying child support, job, time spent with the child) would help your case. Joint custody, however, is not just for the purpose of reducing child support; infact the child support you pay is nowhere near what it costs to raise a child. Consider the ramifications of your sharing custody--what is in the best interests of the child??