You can, and on walls away from water it can last several years if the prep is done properly. On a shower floor or the wet walls of a shower it will fail, usually within a year, and it is not worth doing.
The honest split
This question has two different answers depending on where the tile is, and most advice online gives one answer for both.
- Dry walls — behind a vanity, a dated tile wainscot, a backsplash. Painting works. Done properly it will look good for years and it costs a fraction of re-tiling.
- Inside a shower or tub surround. Do not. Constant water, constant temperature swings, and mechanical scrubbing will find the weakest point in the coating, and once water gets under the film it spreads and peels in sheets.
- Any floor. No. Foot traffic plus water is the worst combination there is for a coating, and a painted floor tile becomes slippery when wet, which is a genuine safety issue in a bathroom.
If you are doing the dry walls, the prep is the job
Ninety percent of failures are prep failures, not product failures.
- Degrease properly. Years of soap, hairspray and skin oils. A strong detergent or TSP substitute, rinsed, then dried.
- Abrade the glaze. Scuff every tile face with 180–220 grit until the shine is gone. This is tedious and it is the step that determines whether the paint stays on.
- Repair the grout first. Paint will not bridge a cracked or missing grout line, it will just crack with it.
- Bonding primer. A proper adhesion primer rated for glossy surfaces — not a general-purpose wall primer.
- Topcoat with a hard enamel or a two-part epoxy, thin coats, full cure time between them. Rushing the cure is the second most common failure.
What it will still look like
Painted tile reads as painted tile. The grout lines fill in slightly and the surface has a uniform sheen that real glazed tile does not. From across the room it is fine. Close up, nobody is fooled.
The honest alternatives
If the tile is sound and you just hate the colour, grout colourant alone transforms a dated bathroom for very little money and does not have a failure mode. If the tile is sound and flat, tiling over it is often possible and lasts properly. If the tile is drummy, cracked, or hiding a water problem, painting it is putting a lid on something that needs opening.
Paint is a reasonable answer to "I need this to look better for two years." It is not an answer to "I want this bathroom sorted."