What could cause a cat's nose to bleed?

Sounds like a good reason to take a visit to the vet.

Probably an allergic reaction
Other than that, I have no clue
Sorry.
Good Luck!
~Loui

If it is coming from the nostril I would say some type of trauma to the body or head.
Is it indoor/outdoor and could have got hit by a car or hurt by another animal?
In any case I would probably seek a vets advice.