OK
I have one POP3 and three IMAP Gmail accounts. The POP3 is fine, all the Gmails stopped working after the 10.10.4 upgrade.
I tried various things suggested above.
The only thing I find works is to go through each of the accounts, in Preferences>accounts>edit the outgoing SMTP server and under advanced, re-enter the correct password.
This works for all three accounts ONCE, and for two of them all of the time.
So, for the one that does not work (its outgoing server now showing Off line), I use one of the other Gmail accounts outgoing servers!
This works a treat.
These **** SMTP / Password issues started with Yosemite. Strangely enough, so did the reordering of email account in the Apple mail account!!!
Spooky or what? If you look at the order in mail vs the order in preferences vs the order in accounts vs the order in the Edit SMTP server list, they are NOT the same.
I wonder if the stupid mail programme is getting its index knickers in a twist and is sending the wrong password to Gmail?
Sort it out Apple
I have only had the Mac a year, transferring from a PC. Now I wis I had just used MS Outlook.
Happy days guys
Tony