Okay, I've been dealing with this for a long while, 9 months maybe?
Sometimes, and I haven't figured out a pattern, when I check mail, it prompts me for the password - sometimes repeatedly!
I've checked the box "save in keychain" thinking it will solve all my woes, but doesn't help.
I guess i should mention that I have BS dsl.
It doesn't do it for all accounts, just about half or one or two. Like I said, it seems random.
Today, it has done it twice on the main account holder

ugh!
Any ideas?