Apparently more Apple users than usual are having trouble with Google's Gmail service over IMAP today, specifically a “mail service imap.gmail.com is not responding" error. Google's aware of the problem and looking into it, and in the meantime have posted some quasi helpful tips on minimizing the chances you Gmail will simply error-out on iOS. Personally, I'd recommend switching to Gmail over Exchange ActiveSync, but if you're a die-hard IMAP lover, you can check them out and hope for the best...
The sad truth is, Gmail's back-end service and Mail.app's front-end client have never worked very well together. Google has incredibly stingy bandwidth and simultaneous connection limits for a provider their size, and Apple is incredibly greedy when it comes to grabbing connections, and downright silly when it comes to re-syncing massive amounts of mail for seemingly no reason.
I've had frequent errors for 3 or 4 years now, and neither company has shown the slightest interest in fixing them. Then, and today, if you use Gmail and have Mail.app open on several machines or set up as IMAP on several iOS devices, you'll have problems. From the likes of Google and Apple, that's simply unacceptable. Gmail needs more reasonable usage limits and Apple needs to be more reasonable about using them.
So while it's all well and good that Google is aware of today's “mail service imap.gmail.com is not responding" problem and are working to fix it, the greater problem needs its fair share of fixing too.
Again, switch to Gmail over Exchange ActiveSync if you can, and if you can't, check out the helpful tips below...
We're aware of a problem where Apple Mail IMAP users are unable to sync with their Gmail account and are seeing a username and password error. We have found that a vast majority of users are exceeding IMAP bandwidth limits and are continuing to investigate.
In the meantime, please review the following suggestions help to reduce the chances of going over bandwidth limits. Avoid these actions that use up a lot of bandwidth: