I did this every time when the problem happens:

https://accounts.google.com/DisplayUnlockCaptcha

• problem sorted in my case!

Google has been so kind as to list all the potential problems and fixes for us. Although I recommend trying the less secure apps setting. Be sure you are applying these to the correct account.

• If you've turned on 2-Step Verification for your account, you might need to enter an App password instead of your regular password.
• Sign in to your account from the web version of Gmail at https://mail.google.com. Once you’re signed in, try signing in
  to the mail app again.
• Visit http://www.google.com/accounts/DisplayUnlockCaptcha and sign in with your Gmail username and password. If asked, enter the
  letters in the distorted picture.
• Your app might not support the latest security standards. Try changing a few settings to allow less secure apps access to your account.
• Make sure your mail app isn't set to check for new email too often. If your mail app checks for new messages more than once every 10
  minutes, the app’s access to your account could be blocked.

Thanks.

Bobby Joseph [iWesabe]