One problem which happens particularly in the O-235 in the C152 and PA38 is the starter spins the motor over fast enough to disengage the magneto impulse coupling but not quick enough for it to generate a big enough spark to start the engine.
When you release the key from 'start' it naturally goes to both (ie 'on'), as the motor slows down now that the starter isn't driving it, the impulse coupling engages and hey presto, the engine starts.
Various things were tried with the starter (to turn the motor over slower) and the magneto lag angle. of course if you mix and match mags and starters....!
Individual engines do behave differently and if you get one with a weak impulse coupling then you're on a hiding to nothing.
All part of the fun of flying (or trying to go flying)....!