Welcome to Apple Support Communities.
You're talking about an email account password being rejected, right?
Have you tried clicking 'Cancel'?
Then go to Mail, Mailbox, Get new mail, and select the problem account.
It will probably work (that is, opening and checking for new mail) automatically without re-entering your password.
Occasionally an important bit or two get dropped as your device passes your information to your intended destination (and return traffic) among the trillions of bits and bytes flying across the internet at any given moment, and things just don't work properly.
Retrying never seems to cure the problem with Apple Mail, but cancelling and reconnecting seems to do it every time.
I encounter this rarely with my .mac email account, but every couple of months like clockwork, several of my other email accounts 'act up' for a couple of days with this behavior, and then are fine until the next episode.
I'm having this problem also. I have not moved, or even tried to move, to iCloud. The problem is affecting all of my email accounts -- MobileMe, AOL, ATT/Yahoo etc. These are all IMAP accounts. I doubt the problem is caused by my service providers since it affects all of my accounts and began at the same time for all of my mail services.
Like JBILDERBACK, I am being prompted over and over to enter my password (for each account) when I try to access my mail. I have put the correct password in each time, but the dialog box keeps popping up again. I have “save password in Keychain” checked.
The problem may be connected with (though not necessarily caused by) the “Account” settings in Mail not being able to retain my password after I have entered and tried to save it. When I try to save the password to Accounts, I am prompted yet again by the pop-up box asking me to enter my password.
I have quit and restarted mail several times; run Disk Utility to repair permissions; rebooted; gone into Keychain to make sure all passwords are correct and Mail has access to saved passwords; and synced Mail Accounts with earlier synced data on MobileMe to see if previous settings can be retrieved.
Same problem: I have a second computer which is running Lion and all's well in the Mail department; the Snow Leopard computer's Mac Mailbox is acting the way described by all above--repeated prompting for password, little triangle w/ "!" in center. I tried the suggestion of cancelling the password prompt window and the first time I did that my mailbox on Mail > Mailbox > Get New Mail was desensitized (greyed out), so I couldn't even select it. Now no longer greyed out, my mac/me mailbox still won't go Online. My other email accounts--Netscape, Gmail--are online and working fine.
I noticed in keychain that my MobileMe Sharing Certificate (not sure what that's for) has expired. Does that relate to my problem?
My Sharing Certificate expired a couple of months ago, while I was still out of town. Have been back home nearly 2 weeks but Mail problems on the Snow Leopard iMac have prevented me from picking up any mail to my Apple mailbox since my return. Inboxes on my other non-Apple accounts are being populated but I can't send anything out.
Kostby: the Cancel out didn't work. Any other ideas? Anybody?