You may have one of two problems. 1. Your roof may be leaking where the bathroom vent exits, or 2. if you are in a cold climate, you may be getting excessive condensation. You need to check the vent pipe and make sure that is is running directly to the vent with no sags along the way. If there are sags in the ducting, condensation will accumulate and eventually run back into the exhaust fan. You can check this by going into the attic and inspecting the ducting. If it is sagging, gently lift the sag to feel if it has water in it.

By chance, is this a first floor bath, with a second floor bath above it? I've seen a case where the second floor toilet overflowed and caused the light fixture in the ceiling below to get wet.