Since 10.11.1 Mail spends most of the time unable to connect to servers
Check to see if you have _mbsetupuser running. If you do your update to 10.11.1 (or to 10.11.2 or 10.11.3, for that matter) has not completed. 

Though the "About This Mac" dialog may say that your version is 10.11.1, you are in fact effectively between versions, and will get all kinds of odd behavior (repeated requests to unlock the Keychain, connectivity problems, wifi issues, etc.) until you complete the installation process, which you should be able to accomplish with a reboot.