Just made this account to post my experience for what it's worth.
Bought a refurbished MacBook on Monday, received it Tuesday. Could not get past the sign in screen to set it up. Skipped that step and let the rest of the set up go forward. It's running 10.11.3. Can't log in to the app store to download the update or any of my other software. Can't log in to iTunes or iMessage or Keychain. iCloud is otherwise fine and syncs right up with my calendar info, etc. I can log into iCloud on the web, and I have no problems with my iPad or iPhone. I called tech support last night (Tuesday) and after an hour of "try this, try that" and no success we thought it might be an overloaded server so soon after Monday's Apple Event.
Today (Wednesday) I brought it to the Apple Genius Bar. We spent two hours trying everything, including a complete wipe and reinstall of the software. Then we exchanged my MacBook for a new one, and tried to set that up, and I couldn't log in on that either. So the problem is with the Apple ID on Apple's servers and not with the computer. I took my original computer back and went home, having set up a phone call with an Apple Advisor for this evening who would be able to help me they said.
On the phone tonight I spent an hour trying various things with the Apple advisor who did a screenshare with me this evening. She escalated me up to a Senior Advisor, who broke the sad news that there is nothing to be done but to wait for Apple's engineers to fix the server problem. The Senior Advisor said there was a memo out to the tech support people from the engineering department acknowledging that there was an issue. He said the fix should be in a matter of days not weeks. My only option is to keep trying to log in until it works. He was apologetic and certainly every one I've dealt with at Apple Support has been courteous and kind, but that is what I was told so I figured I would share it here.
Actually typing this all out for other people with the same issue has been the most calming aspect of a frustrating experience. Fingers crossed for a quick fix.