I was getting repeated prompts for Keychain passwords on my Mac, and repairing Keychain Access, which Red Sweater had promoted me to try earlier (04 Nov 2015), wasn’t working anymore. I even tried resetting Keychain but no joy either.
I sniffed around and fingered the culprit as Juniper Pulse, my VPN for NUS access as the culprit. Indeed it had beach-balled when I tried to initiate a connection.
Earlier, I had updated Juniper Pulse to what was now Pulse Secure version 5.1 (04 Nov 2015) from the installer on the NUS Comcen link. This worked on OS X El Capitan (10.11). I had to take the additional step of delete the existing NUS VPN profile, restarting my mac, and adding a new profile.
Early this morning (14 Nov 2015), the installer on NUS Comcen was now Pulse Secure version 5.1.revision6 build 61491 (5.1r6.0-b61491). Aha – a new version! So I installed it and replaced the profile, and my problems with Keychain were over.