I was hunting with several friends up in the Yukon 3 years ago. We shot a pretty big moose on the second morning and had to hang it up in a tree. I set up a rope with cans attached as a sort of primitive warning bell as there were bear tracks all over and I was being cautious.
I had a side by side coach gun with solid copper slugs (just for camp protection) and pulled it out at 5:30 am when the cans went all crazy clattering. The other 2 guys heard it also and loaded their rifles up. It was still pretty dark, just enough light to see about 40-50 feet. I hopped up on top of the Camper and my buddies stood back to back.
Then without warning , a grizz came out from the brush but was too close to the camper for me to get a shot. My friends were about 30 feet away and one let off a .375 H & H 270 grain bullet that entered the vitals just behind the shoulder. The grizz got knocked over, didn't move for about 2 seconds , then got up and bolted towards him. He had another round chambered by then and hit him in the left shoulder. The bear took off into the bushes.
I got off the camper, and the guys said we should let him be for a few minutes before going in after him. I said "lets let him be until its brighter out you idiots - or you're goin in after him yourselves!!!!" They woke up to their senses and we waited. About 45 minutes after, it was bright enough out and I took my .300 win mag with me, my friend took the .375 open sights, and my 2nd friend took his .338 win mag. We got a blood trail after 5 minutes and carefully and slowly followed it until we saw the bear laying upright against a tree , kind of "hugging" it. It was still alive so my friend who originally shot it, immediately hit it in the back of the head to end its misery.
I have to say I was never more "scared" in my life, as I went into the brush under protest, but also they made me feel guilty in case something happened to them. My friend also said that he had the truck keys in his pocket, and I would have to get it from the bears stomach if they got eaten and I didn't come. So the moral is that as stated before, he got rocked with 2 solid shots and still took off running. He made it about 250 yards.
My Dad always said that bears are made different, and they are sometimes too stubborn to believe they are dead. They can be fatally shot, and still have enough adrenaline to make a run on you or AT YOU before the heart and brain physically shut them down.