If your iPhone/iPad/iPod Touch Mail app is generating these errors despite connecting to the same server without difficulty prior, a quick fix is to:
- Double tap your home icon while on the home screen to bring up a list of all running apps
- Press and hold on the Mail icon until a red minus appears, press that minus
This will quit that currently running app (whereas pressing the home icon normally simply switches GUI control to a new process – i.e. the app is left to run in the background). Once the app has been quit and relaunched these errors should go away.
I can’t substantiate this, this is just a theory, but I suspect that since the iOS Mail client (and OS X Mail) are not particularly good about limiting the number of connections to an IMAP server to sane and reasonable numbers that these errors are generated when no more connections are permitted, and will go away on their own after unused connections timeout. Quitting the app presumably kills these IMAP connections in IMAP IDLE state.