Do black holes ever get full?

No, black holes never get full.

The size of a black hole is fixed by the amount of mass it contains. For each increment of mass the size of the Sun, its radius is about 2.6 kilometers, so if a black hole has a mass of 10 times the Sun, its radius is 26 kilometers. The more matter that falls into it, the bigger it gets, so in fact it never fills up inside.