Damn... well, there you go. My Hardcore Barbarian finally dies right in Act 1 of Nightmare, just before I entered Leoric's Manor.
I encountered TWO Elite Packs in this area where I used up all my offensive skills to dispatch the first one. I didn't see the second pack until it was almost on top of me. With my offensive skills still on cooldown I was caught vulnerable, pinned by the Elite's jailer affix, and unable to run.
I didn't feel as bad as I thought I would but I have to be honest and say I didn't expect to die so soon (not after breezing through Normal with very little difficulty). It was my mistake. I got complacent. I wish I could say it won't happen again as I won't be attempting Hardcore again.
One thing this has taught me though, is how strong Barbarians really are. My first Diablo 3 character was a non-hardcore Barbarian but, because I had a hard time getting a grasp of the game that early, I ditched that character at Level 16 and went with a Monk. And it was with a Monk that went the distance until Level 60.
But now I've gotten a better handle on the game and a better handle on Barbarians that I've revisited my Level 16 Barbarian and I'll be taking him from there. It will be the character I will be playing for much of the time from now on.