Mail.app stops Sending / Receiving

This has been killing me for a few weeks. I got the following from Apple help and the part that worked was the last step, changing the Authentication to password instead of token. Hope this helps. BTW you cannot change your incoming mail server since that is greyed out.

From the Mail menu, choose Preferences.
In the Preferences window, click the Accounts tab if it is not already selected.
In the Accounts list, select your iCloud email address.
Click the Account Information tab.
Verify your SMTP server settings with the following information:
Incoming Mail Server: imap.mail.me.com
User Name: Your iCloud email address
Password: Your iCloud password
Click the Advanced tab and verify the following additional settings:
Port: 993
Use Secure Sockets Layer (SSL): Should be enabled
Authentication: Password

I've had the same problem and appear to have solved it (although it is early days yet). I closed mail then deleted the hidden .offlinecache folder found here:

/Users/theuser/Library/Mail/V2/IMAP-imapfolder

When mail was reopened it worked fine and is still working.

Maybe this is an answer?

Struggled with this problem for a while. Could receive IMAP OK using MacMail but sending would not work. The attempts resulted in a local reject stating that the sending account was offline.

My very able ISP provided the solution, for my problem anyway. She said that this was a common problem and that the OSX default setting for the port number simply doesn't work properly. It is important to specify one specific SMTP port, not to rely upon the defaults. We did this and it all worked just fine. Hope this helps somebody else ....