The swimming pool test
There are two ways to not be able to swim: you never learned (no sé nadar) or you're injured/the pool is closed (no puedo nadar). This single example captures the entire distinction.
Poder in the conditional for politeness
¿Podría hablar con el director? (Could I speak with the manager?) The conditional of poder is one of the most important politeness tools in Spanish — softer and more formal than the present.
May vs might: puede que + subjunctive
Expressing probability in Spanish uses poder: Puede que llegue tarde (He may arrive late). Note that puede que always triggers the subjunctive.