I am having this problem and my parents are having it also. Since we are all having the same problem, and since it started around the same time for all of us, without changing things on our end, it is pretty obvious to me that it is a Verizon problem. I am a software engineer and this is my theory:
Verizon's servers started to get a bit "flaky" last week. All mail applications periodically check the servers to see if there is any new mail. When the server is acting funny and the connection is rejected when checking for mail, most mail applications probably wait and try again a little bit later. But Mac Mail probably assumes that it is getting rejected because of a bad password, so it tries to get the correct password from the user, even though it already has the correct password. In other words, the problem is there for everyone, but those of us using Mac Mail actually see the problem.
Again, that's just a theory.
So, I don't think it's any settings on our end. And from what I've heard, trying to get Verizon to give support for a Mac is almost impossible. Since this problem isn't noticable except when using Mac Mail, I'm sure they would try to blame the Mac and say it's not on their end. I've called Verizon for support in the past, and there is no way I can deal with the frustration of calling them about this one when I know it's their problem and not mine.
The only thing you can really do is to increase the amount of time between mail-checks in Mac Mail. Then it won't bother you as much. You can always hit the "Get Mail" button to check for mail in between the automatic checks. Hopefully they eventually get their server problems fixed and then you can let it check for mail more often again.
I am curious where everyone is from. Maybe it is a problem with local servers? I am from Pittsburgh.