I remember reading that it can be an issue with kesubos and gittin (if a name isn’t used, a) what happens if it’s left off the kesubah/get and b) is it really a name?), but I don’t think there are any actual issues with it.
If you’re allowed to add a name to a choleh then presumably the number of names you have isn’t a big issue (though I’m completely extrapolating on my own here).