Using iOS 9.0.1 and the native Mail app, downloading mail from Google mail stops working after about a week. The message the system displays is that it is not able to connect to imap.google.com server, and to check the password. Turning off and then on mail for the account does not resolve the problem. Neither does just quitting Mail. Resolving to removing and then adding again the account is one way to do this.

Another way I was able to figure our, without removing the account from the devices, is the following sequence:

1. Go in Settings and turn off mail for the account.
2. Quit Settings.
3. Quit Mail.
4. Go in Settings and turn on mail for the account.
5. Open Mail.

The above bug and fix I have confirmed on an iPhone 5 (Verizon) and iPhone 6 (ATT).