Both Mail and the iPhone initially reported that either my password or username was incorrect, but after restarting my MacBook Air, the message changed to what you see above. While I had been sending some large files through email last week, I wasn't using anywhere close to the 2 or so GB per day of bandwidth that Google allows.
A quick search revealed that there's a known bug where accounts are locked out for exceeding IMAP bandwidth limits when using Apple Mail under Snow Leopard. To quote Google's support page,
This is a known issue for users of Apple Mail on the Snow Leopard OS. We are working with Apple on a resolution to the issue. In the meantime, there are a few workarounds to reduce the likelihood of hitting the bandwidth limit:
For my business, I keep my email client up and running most of the day, so the third workaround was not an option. Syncing attachments is easy to turn off in Apple Mail Preferences for Gmail IMAP accounts by selecting "All messages, but omit attachments" under "Keep copies of messages for offline viewing" on the Advanced tab, but it didn't resolve the problem.
- Do not sync attachments.
- Reduce the number of folders you sync by using the Advanced IMAP Controls tab in Gmail.
- Close your email client when you are not using it.
I fortuitously noticed that my MacBook Air seemed to be losing a lot of disk space, and searching the ~/Library/Mail folder indicated that the issue seemed to be in the "Recovered Items" Inbox for this Gmail account -- something I didn't even know existed! It was taking up a whopping 18.66 GB of space, so I went into Mail and deleted the "Recovered Items" inbox.
While this didn't fix the issue immediately, it did eventually allow the account to come back on line with no problems. All told, I was without Mail.app and iPhone access to the account for four days, but at least I could check my Web mail occasionally to see what was coming into the Gmail account.
If you're running Snow Leopard and get bit by the "bandwidth bug" with a Gmail account, I hope that this tip can help you out.