I'm interested both in your personal opinions and in information how this problem is dealt with in your countries. Lest you get a wrong idea: It's not a personal problem for me.
(German) Scenario: Woman gets child, one man gets the idea he is the father of the child. He acknowledges being the father and so officially he is and pays. Additional information (his doctor telling him that his sperm count is extremely low, rumours about a love affair of the mother of the kid at the time of conception, relatives shaking their heads in disbelief about the lack of similarity,...) makes him doubt his fatherhood at a later time.
Often, at this time, he is no longer with the mother of the child who has custody rights. But he pays for the child. At this moment, he thinks how good it were (for his finances) if another man had to pay. He asks for a paternity test. His former friend/wife who has the custody rights says no to that for she is the only one to decide what is best for the child. The man goes to a lawyer who tells him that in Germany it is nearly impossible to sue for a paternity test against the will of the child (which is devined by the mother).
At the next visit of the child the father gives her a chewing gum which she chews and spits later into the bin. The hopeful non-father fetches the used gum from the bin and a few days later he knows his hopes of unfatherhood are fulfilled. Paternity is impossible.
He walks again to his lawer who tells him that knowing definitely from a surrepetitious test is not a valid reason to damand an official paternity test against the will of the mother. In addition to that he may even be punished for making the test. He asks in disbelief: One would believe me when I say I made a surrepetious test and punish me for that but one would not admit the test result as an indication of doubt about the parenthood? Yes, says the lawyer.
One case has now reached our highest court: A man had acknowledged paternity. Later he was told by his doctor that his probability of fathering a child was a mere 10%. With that knowledge he asked for a paternity test which was not granted by a court. A surrepetitious test (chewing gum) confirmed what he had already suspected. Paternity impossible. This sure knowledge may not be used and so he still has to pay.
(End of scenario and case)
(1) I am in general against surrepetitious DNA tests (imagine the cup of coffee you had at a job interview is tested for your cancer risk), but I can understand the motivation of a doubting father. (2) As a scientist, the idea of not using sure knowledge sounds nonsensical. (Some old psychological experiment would not be allowed today for ethical considerations, but the results are still in the books) (3) A minor problem (for jurists): If someone spits out a chewing gum isn't she giving up at that moment all rights she had in relation to that gum?