Only problem is, it would be impossible to tell which child came from which parent. Indeed it's impossible to tell if both had the same father or if there were one from each. So I can't see any plotline that works except the 'we don't know who the father is' one.
Actually, sequencing the entire genome of both possible fathers, looking for (very rare) mutations could enable the parents to be told apart genetically. However, the first human genome sequence (the only one we've got) cost several billion quid and took 10 years, although it could probably be done in three now. Unlikely for a paternity case.