All of a sudden, I am being frequently asked to sign in with my Apple ID to my iTunes account. A third-party calendar app, BusyCal, asks me the same, to sync events with iCloud, but now refuses to connect despite repeated entries of my valid password. Today, and last night, iCloud/iTunes insisted I answer my security questions in order to Unlock my Apple login. I am annoyed by this behavior and don't know what has set it off. Any ideas? MBPro, 10.12.3, and also on two iOS 10.2.1 devices

Chuck, it seems to happen when they are working on the servers. I went through it a couple of weeks ago. I run into it every few months. My headache this week is with my gmail accounts constantly asking for passwords. When the Apple and Google servers are slow to respond, I get the password request instead of whatever I'm using saying it's not responding.

Followup:
This problem seems to have magically gone away now, so Phyllis may have had the answer.
Also, the BusyCal apps were just updated, and said to fix, among other things, problems connecting to and syncing with iCloud.