I flew BA and bought the smallest recommended size crate for my cat, and at check-in they made my mom go to the pet store and buy the next size up (which gave him a whole 1.5" extra of headroom, and he was already pretty small compared to the crate, so I'm not sure what the point was!).
I think the smallest one they recommend is 21x16x15, and the next size up (that they allowed him to fly in) is 23x17x16. He's a pretty average sized cat (frame-wise) and 13lbs.