How fallible are DNA paternity tests?
Discounting lab error and other kinds of human incompetence, there aren't very many things that can confound a paternity test. Normal DNA damage cannot, but there are kinds of genetic chimeras that get strange results.

Normal human DNA damage can't add up fast enough to change the results without killing you several times over, but there are a surprisingly large number of people who are chimeras. 

The relevant particular flavor of chimerism is tetragametic chimerism, where essentially parts of your body are sibling-related to the rest. This occasionally is discovered during paternity testing, in one vivid instance when a woman was provably not the mother of the newborn that she had just given birth to. Technically I suppose the baby was her skin's niece.

The incidence of this was thought to be very low, but it's almost impossible to detect without DNA testing. We know the incidence is much higher than we thought, but we don't know how high.