Why a leaking ceiling is the landlord’s job
Yes, usually. A leaking ceiling is generally the landlord's responsibility, because it almost always comes from a roof leak or a plumbing problem above — both part of the structure the landlord must maintain.
Water dripping from or staining a ceiling points to something above the unit: a roof that is not keeping out rain, a leaking pipe, or an overflow from an upstairs apartment. Keeping the roof watertight and the plumbing in good repair is the landlord's duty, so fixing a leaking ceiling — and repairing the water damage it causes — usually falls to them.
Painting over a water stain without finding and fixing the source is not a real repair; the leak simply returns. If your landlord treats a leaking ceiling as a cosmetic issue instead of finding the cause, the problem will keep coming back. A free review can help you understand what your landlord is actually required to do.
This is general information about California tenant rights, not legal advice. Every situation is different.