Family Sharing, this action cannot be completed at this time

I had this problem too: I had been sharing Apple IDs for the App Store between devices, and was trying to set it up the right way with separate IDs and Family Sharing. But this non-helpful error message kept popping up.

Eventually I solved it by logging out of the main, App-Store Apple ID on the device to be changed before attempting the family sharing, then everything works fine (I was hesitant to do that originally, because I was afraid it would want to uninstall apps, the way turning off iCloud wants to remove shared items).

My device was being backed up to the computer instead of iCloud. I found that once I changed the setting so that it was backing up to iCloud I was able to proceed successfully and complete Family Sharing.

Nope.. my backup is to iCloud.. and I too get this message on my wife's phone..