Since 10.11.1 Mail spends most of the time unable to connect to servers After years of working smoothly, through many OS updates, including 10.11.0, my Apple Mail client now only succeeds in connecting to servers intermittently since updating to 10.11.1. I'd estimate that on a typical day, it manages to work about 10% of the time it is running. Mostly I just quit and leave the app closed to avoid the annoyance of constant warning dialogs interrupting me while I'm working. (Connection Doctor never shows any errors or difficulties.)

Is there something simple I might do to fix this? Is this a known problem with 10.11.1?
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.