POP Connections in Mavericks Mail have Become Intermittent

Question

New iMac upgraded to Mavericks about 3 weeks ago. Mail worked just fine and sending mail with SMTP still works OK. However...
Over the last three days Mail started to occasionally fail to pull in new email. Now it only fetches new mail whenever it feels like it -- 2 or 3 times a day. The rest of the time it sits there with the little error/alert icon on the Inbox line that usually leads to a message indicating the server connection has failed -- The server returned the error: The connection to the server “mail.mediacombb.net” on port 110 timed out.  Once in a while we get a more elaborate message that includes a button for the Mail Connection Doctor and another that is supposed to bring all accounts back on-line. The "Doctor" always finds that there is a failure of the POP connection.
We have tried alternate mail ports -- with and without SSL (as suggested by our mail host Mediacom) and nothing makes a difference.
Heard that killing our Mail account and reestablishing it from scratch might fix this problem. BUT, part way through this process, I came to a warning that doing so would cause us to lose not only all of our preferences and other settings, but would also result in the loss of all of our mail messages!! Can't afford to do this.
If we do wipe out our mail account and all the locally stored messages are lost, can the previous (lost) mailbox contents be recovered -- perhaps from a recent backup? If so, how to do this?
Any other suggestions?
Thanks for any help!

Answer 1

The copy to the Desktop was most likely a backup in case something went wrong.
You might want to download the combo updater and refresh your files.
OS X Mavericks 10.9.3 (Update (Combo)
http://support.apple.com/kb/DL1746
MORE INFO ON WHY RUNNING COMBO FIXES ISSUES
Apple updates available from the Software Update application are incremental updates. Delta updates are also incremental updates and are available from Apple Downloads (software updates are generally smaller than delta updates). The Combo updates contain all incremental updates and will update files that could have become corrupted.
Combo updaters will install on the same version as they're applying--no need to roll back or do a clean install. So if you think you've got a borked 10.8.4 install from a regular update, just run the 10.8.4 Combo Updater on that system.
"Delta" updaters can only take you from one version to the next. For example: 10.9.1 to 10.9.2. If somehow the 10.9.2 is missing something it should have, and that something isn't changed between 10.9.1 and 10.9.2 it will still be stale after the delta update.