Since I posted this, there has been more, but slightly different, weirdness.
In the middle of a session yesterday, Apple Mail started to do something different. When I typed in the password, it briefly worked (downloading one or two messages) and then the error dialogue box popped up again so clearly the password was always correct.
A short while later, it accepted the password and stopped complaining until ten minutes ago. This time I only had to type in the password once and it is working at the moment but who knows how long for?
I've been having the same problem, but only on MacBook Air. Meanwhile, my imac, ipad and iphone are working just fine.
I was on the phone with Apple yesterday for about an hour and there was still no resolution.
I'm ready to throw the computer across the room.
However, I'm going to download Mountain Lion in hopes that that fixes the problem.
Has anyone found any resolutions since the original post?