Consistently get "server imap.gmail.com is not responding" in Mail
I keep having issues with receiving mail from my Google Apps accounts while using the Mail app on my iPhone. I currently have one Google Apps account that houses two different domain names so I can have separate emails for each domain. Up until very recently, this setup worked perfectly without issue for the past year or so, but suddenly this started happening. Usually, one of these emails will be working fine while the other is the one that stopped working. But I've noticed it's not always the same email that's having the issue, as sometimes it's one and sometimes it's the other. And I can't tell if it's an issue with Google or with Apple, but I ONLY have the problem in Mail on my iPhone. In any other email app or in Safari on my iPhone, it works fine. On my desktop in Apple Mail or in the browser, works fine. The exact error message states "Cannot Get Mail: The mail server 'imap.gmail.com' is not responding. Verify that you have entered the correct account info in Mail settings." The only way I can get it working again is if I delete the account from Mail and then add it again. Suddenly it starts working again using all the same settings I used previously...... Until the next day, suddenly it breaks overnight. I've noticed it's consistently happening in the very early hours of the morning while I'm asleep. I'll wake up and find the error again. Right now this has been a daily affair, and is proving to be really frustrating. For the record, this is on my iPhone 6s with iOS 9.0.2, and the issue is present on both Wi-Fi and LTE. Any ideas?
iPhone 6s, iOS 9.0.2