Probably going to write this wrong, so I apologize. If the identified father had brothers, it's possible one of them could have been his biological father instead, no matter what the birth certificate says. I don't think they're disputing a familial relationship, only that their father is his father. That's how I interpreted it.
For example, in the Allenstown NH case, they narrowed down the possible father of Denise's daughter, Dawn, to a family of brothers. IIRC, all were married at the time she was conceived, and all the brothers deny it, but, based on DNA, one of them did.
Then there is a news reporter whose mother was given up for adoption. So, he went searching for his biological maternal grandmother (I think he found the grandfather). It came down to four daughters, all who had passed by then, and no one still alive knew which daughter had a baby and gave it up for adoption. He never found out who she was, only that she was one of four.