Can someone sue for back child support without ever granting a test to prove paternity before?

If his name is on the birth certificate they can. He would have to prove he isn't the father and fight it if he thinks he is not the father.

Can someone sue for back child support without ever granting a test to prove paternity before?

I was older when my mother told my biological father of my existence. The state would only go back so many years and enforce child support on him. He wasn't on birth certificate and dna test hadn't been done. It was a one night deal and he never knew until she told him. So, I think it would depend on whether the father knew or not.

Would also depend on the state and the overall situation in my opinion.

If his name is on the birth certificate, yes.