okay well they got me to take out some .plist files to see if that would help it didn't. I'm not sure what ones that they deleted for my self maybe someone here would be abel to tell you what ones to take out but didn't work the one that came close to working was this going into keychain and deleting the account(s) then reading them again.
if you are to do this please back up first with Time Machine. You can do this with a usb drive.
Okay so go back to where you had internet accounts and make sure mail is shut. click the - on the account your having problems with then quit system preferences.
Okay so then press cmd and space bar at the same time then type 'keychain access' into the search then click on 'keychain access'
once it is open go to the top of the window and put your email address that you are having problems with.
once you have put in your full email, have a look at what comes up i deleted things in the kind colome with 'internet password' and anything with under name that had smtp.gmail.com or imap.gmail.com so it would look like this, map.gmail.com - internet password.
right click and delete them. make sure not to touch any other files that dont say them things on them.
Once you have done this quit keychain access and then restart the computer, once the computer is restarted go back to system preferences and internet accounts and re-add the account.
Hope this helps. if something is not clear just ask.