When I was at school, I wasn't taught to swim (or ride a bike). There were swimming sessions during games periods, but no effort was made to teach those who couldn't swim to do so - the teacher just left us non-swimmers in the shallow end with floats and stuff while he did water polo with those who could swim.
(I seem to remember a similar approach to most sports, actually - no actual teaching took place, just playing. If you didn't know the rules, tough).
I learned to swim pretty much by myself at about 14. I'm still not a great swimmer, and I don't really enjoy swimming much unless it's really hot, but I can swim a couple of hundred yards and at least I don't drown!
I only learned to ride a bike when I was nine - didn't have a bike until then.
I've had a student who couldn't ride a bike. He learned while he was with us.