I have similar experience to FrustratedCinOK.

I have one Gmail account set up in the mail app on my iPhone, I do not access this account via a client app from any other device, I do check it via Chrome and Safari browsers, those work fine.

Email on iPhone worked great for close to a year. A couple days ago suddenly started getting the "Cannot get Mail - the mail server imap.gmail.com is not responding....". I had made no changes to my phone. I have tried following the instructions listed in TS3058, including trying the gmail URL (also see no captcha) and deleting the account from my phone, nothing has resolved this issue.

The iPhone account is setup to also sync my Google Calendar, this work fine.

Running 5.1.1

Any assistance would be greatly appreciated.