My experience was just the same at the genius bar, we tried everything to get a 3 day old mini 3 working - the assistant tried her two AppleIDs one worked one didn't! Must have created about 4 different AppleIDs to see if that made a difference - but it didn't. We tried one of the new AppleIDs on a store iPad (running iOS8.2) - and it validated OK - so we were left to summised it was an iOS8.3 problem. So left the Apple store with a replacement mini 3 running iOS8.0 on which the AppleID worked. Got it home - and 24 hours later after backing up on iTunes it fell over. Calls and escalation to engineering and a week later fiddling with the clock timing and suddenly it worked, and has carried on working. Still not brave enough to load 8.3 though! So keep up the calls to AppleCare - good luck !! and enjoy following the discussion above (48,000 views must mean something!)
mini 3 iOS8.0