Mavericks Mail Inexplicably Stops Retrieving Email

Question

It doesn't matter if I set the preferences to check automatically, every 1 minute, every 5 minutes, etc. Mail just stops retrieving, and even more aggravating is the fact that the Activity monitor runs at the specified interval, implying that my mail is being checked. And yet, if not for checking my email on my iPhone, I'd think everything was fine, it's just a particularly slow day. When I see that in fact I've gotten a number of emails, I check my Mavericks Mail manually, and still nothing comes in. It's only upon deactivating all my accounts and then reactivating, or quitting the app and then restarting that I get all the mail I've already gotten on my iPhone. This is pretty infuriating, as I get caught up in my work and forget that my Mac Mail is "moody."
I'm running Mavericks 10.9.4 and Mail 7.3.

Answer 1

You may be affected by the bug, but this is the first time I've heard that it affects some accounts and not others.
Please read this whole message before doing anything.
This procedure is a test, not a solution. Don’t be disappointed when you find that nothing has changed after you complete it.
The purpose of this step is to determine whether the problem is localized to your user account.
Enable guest logins* and log in as Guest. Don't use the Safari-only “Guest User” login created by “Find My Mac.”
While logged in as Guest, you won’t have access to any of your documents or settings. Applications will behave as if you were running them for the first time. Don’t be alarmed by this behavior; it’s normal. If you need any passwords or other personal data in order to complete the test, memorize, print, or write them down before you begin.
Set up one of the affected accounts while logged in as Guest. Test. Don't open any messages. Just see whether they're retrieved.
After testing, log out of the guest account and, in your own account, disable it if you wish. Any files you created in the guest account will be deleted automatically when you log out of it.
*Note: If you’ve activated “Find My Mac” or FileVault, then you can’t enable the Guest account. The “Guest User” login created by “Find My Mac” is not the same. Create a new account in which to test, and delete it, including its home folder, after testing.
Step 2
The purpose of this step is to determine whether the problem is caused by third-party system modifications that load automatically at startup or login, by a peripheral device, by a font conflict, or by corruption of the file system or of certain system caches.
Please take this step regardless of the results of Step 1.
Disconnect all wired peripherals except those needed for the test, and remove all aftermarket expansion cards, if applicable. Start up in safe mode and log in to the account with the problem. You must hold down the shift key twice: once when you turn on the computer, and again when you log in.
Note: If FileVault is enabled, or if a firmware password is set, or if the startup volume is a software RAID, you can’t do this. Ask for further instructions.
Safe mode is much slower to start up and run than normal, with limited graphics performance, and some things won’t work at all, including sound output and Wi-Fi on certain models. The next normal startup may also be somewhat slow.
The login screen appears even if you usually log in automatically. You must know your login password in order to log in. If you’ve forgotten the password, you will need to reset it before you begin.
Test while in safe mode. Same problem?
After testing, restart as usual (not in safe mode) and verify that you still have the problem. Post the results of Steps 1 and 2.