If the threads are not actually damaged (as in only poorly cut from the factory or heavily clogged) then running in a proper piloted tap won't really remove much material. This is called chasing the threads and one can do this many times to a BB shell (if they know how to). However if there is actual damage then the taps will tend to remove the material where it shouldn't be and it's this removal that one has to be very careful about because it's a lot more involved to replace that material
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Thie situation is one that can go well or really bad dependent on the person doing the work. I don't think learning how to chase threads is best done long distance. If the cup won't thread in with mild wrench effort I suggest taking the frame to a shop that knows about this stuff and has the correct tools. Andy