Mail cannot connect to google account, won't open once quit Pretty simple - mail showed as unable to check for new message (tilde by the inbox, etc.). The activity doctor confirmed it could connect to the internet but not the mail servers. Quitting mail and trying to re-open it resulted in the error "Mail cannot save information about your mailboxes because of a network problem."
I had this problem yesterday and restarting seemed to fix the issue. When it came back today I looked online but couldn't find any answers. A look in the console showed that the issue was not network related, but actually caused by low disk space. Sure enough, making some space on my drive ( . . .was down to the bytes) immediately fixed the issue.

Moral of the story: Keep your disk clean!