MacOS – Mail.app encounters recurring connection error (to Gmail IMAP server) after installing Mountain Lion

Follow these steps:

1. Open Mail and delete each of the conflicting accounts. Close Mail.
2. Go to ~/Library/Mail/ and make sure any IMAP-username123@imap.gmail.com folder has been deleted. If there is any such folder left, delete it now.
3. Open Mail and create one of the accounts and let Mail load your messages. Close Mail.
4. Go to ~/Library/Mail/ and verify that a folder called IMAP-username123@imap.gmail.com has been created.
5. Open ~/Library/Preferences/com.apple.mail.plist. If you have installed Xcode, this file will open with Property List Editor. Otherwise you can download a free trial of PlistEdit Pro to open this file. Using a plain text editor with .plist files is more difficult.
6. With the .plist file open, click the disclosure triangle for MailAccounts. Here you will find Item 0, Item 1, etc.
7. If you click the disclosure triangle for each of the Item headings, you will find that one of them has the value username123@gmail.com or username123@company.com for the AccountName key, depending on which account you created in step 3.
8. Change the value for AccountPath for this Item so that the path will not collide with that of the next account you are to create.
9. Save and close the .plist file.
10. Open Mail and let it reload your messages. Close Mail.
11. Go to ~/Library/Mail/ and verify that a folder with the path value you provided in step 8 has been created. Delete the old folder called IMAP-username123@imap.gmail.com.
12. Open Mail and create the remaining account normally and let Mail load your messages.

The canonical fix for troubleshooting Mail on OS X is to run the Connection Doctor (it's slightly buried in the Window menu without a keyboard shortcut to summon it) to isolate whether you have a general network error condition or a specific error on one or more of the accounts.

Each mail account will typically have two servers - different internet ports and potentially different username / password / authentication details.

Once you can isolate a specific error message on a specific port to a specific IP address, you can begin to troubleshoot whether you have a configuration issue, a network reachability issue or a server issue.

The show detail button is helpful to log the exact transactions and communications between Mail and the server.

If you have a vexing problem, I have found that disconnecting the network connection and quitting Mail (or forcing a reboot while the network is off) will let you start Mail and disable all Mail accounts except for one and also check for a stuck message or two in the Outbox. File those away as drafts so that you can then run connection doctor, re-enable the network and systematically work on the communication needed between the Mail program and your mail server.