>Check the GFCI outlets in the bathroom to see if any of them have been tripped. Sometimes those get daisy chained together, even though that's not the proper thing to do.

This in particular is common in homes built between the 60s and the 80s. They used to have a two prong outlet for a plug in electric razor, tied to the bathroom lights but otherwise provided no GFI or even basic grounding protection (they did have a transformer to step down the voltage, so it wasnt a complete death trap). Homeowners and landlords typically just swap the two prong for a GFI receptacle but change nothing else about the circuit, which can take out the lights when the GFI trips. If you have an unusually large cover plate around a bathroom GFI, this has happened in your home