I suspect that in the Harry Potter universe there are degrees of magicalness, not merely yes/no.

In the nearest similar other fictional scenario, the Jedi in Star Wars, their powers are mediated by in-dwelling micro-organists called 'midichlorians'. I do not know whether this is canon or fan-writing, but I read on the WWW that the midichlorian level must be 7 or more to be accepted for Jedi training, and Han Solo's midichlorian level is 4.

Likewise in Harry Potter? Perhaps if someone's magic power level is above zero but is too low, the result is some partial magical ability, like Mrs.Figg's ability to see Dementors, but not enough to be trained at Hogwarts.