The paint in our tub has been developing air pockets under it for a while, and this morning a big chunk ripped off. 
As far as I can see, the layer underneath is undamaged except for the stains. Might it make sense to strip off the rest of the paint and live with the stains until I have the time and money for a better repair?
For those wondering how such a big chunk came off, last night I took off the non-slip mat which was suction-cupped to the tub and scrubbed the tub (non-abrasive cleaner, but rubbing pretty hard). This morning I stepped in for a shower, and the area where the mat had been detached.