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there's some movement/play in the spindle. Would this be because the BB is mismatched?
Quite unlikely, as long as the spindle and cups are both designed for the same ball size and in good condition, you should be able to adjust out any free play. Only weird shit uses anything other than 1/4" balls. The only other thing which can trip you up is not having the two BB shell threads coaxial and perpendicular to the shell faces.
Noob question.
I'm having a few issues fitting a square taper BB and crankset, there's some movement/play in the spindle.
Would this be because the BB is mismatched?