If there are to be doubts they will probably begin before birth - when testing is not practical. Testing after the doubts begin seems to be a pretty likely outcome.

Also, I figure the guy is going to want to be the one who administers any paternity testing. A test administered by the girl leaves some opportunity for deception by switching samples.

Testing to resolve uncertainty over paternity seems like a good case of humans consciously and deliberately caring about the welfare of their genes. Some seem to think that evolution's motivation for a man to reproduce his genes comes is in the form of sexual desire and pleasure - but for many, this is just not so.