I do not know if anyone else still needs help with this topic, but this is what worked for me.
My Specs:
MacOS 10.6
Mail: Version 4.5
My Problem: Password rejected/ not recieving or sending mail.
What I did to solve this was to delete my account and force the mail application to find my information. After it failed to find the info, I set it up manually. If you're willing to get into the nitty gritty and manually do something, then please continue reading.
Steps:
1) Go to mail application and open 'Preferences' in the 'Mail' drop down menu.
2) Go to Accounts. Select the mac account in question and then delete it by using the little minus sign at the bottom left of the window. Click 'Remove'.
3) After deleting the account, press the plus sign beside the minus side to *** a new account.
4) Type in your name, your @me.com email address, and the password associated with the account.
5) It should take a moment and NOT FIND the mail server it is looking for. That is fine, click continue anyways.
6) Now the page lists various information about the INCOMING mail server you are trying to access. This is good. Change 'Account' to 'IMAP'. Change the incoming mail server to 'imap.mail.me.com'. Make sure you add the @me.com to your username. Click continue.
7) Now onto OUTGOING mail server stuff. Change the outgoing mail server to 'smtp.mail.me.com'. By changing this it will delete the setting to use authentication.
8) Click that you want to use authentication. Make sure to add @me.com to your username. Click continue.
If you have followed these steps, it should have set up and accepted your information. Your new email should take a few minutes to arrive but it should arrive. Hope this helps and have a nice day! 🙂