Yesterday, I had two IMAP accounts, for which all of my info was current, disappear from my InBox though they still appear when I go to the Accounts area of preferences. It's not just that the emails themselves disappeared but the names for the accounts themselves are gone.  (Specifically AllerStaff and Leikenfamily @ SBC. I re-created the latter so you can see in the InBox area I have LeikenFamily new but don't want to try with the AllerStaff one. The AllerStaff one didn't just disappear on one computer but on two! (Though the accounts are fine on my iPhone — for now.)

How do I get them back?

I'm on a Mac (one MacBook Pro, one iMac) and they're both running Mavericks. One of the accounts that disappeared was a Gmail, the other was Yahoo/ATT, both IMAP.