I've had three trout take me into backing. One on the Green River in Utah, one on the San Juan River in NM, and the third on the Madison River in Montana. I never landed any of them. Truthfully, I could have had more take me into backing if I had let them run, but I fight fish aggressively. Once a fish gets to the backing, the drag from 90 feet of fly line in the water adds to the tension of the tippet and you probably will never see that fish again unless you are a Gary Borger.
I was with Gary Borger when a trout took him into backing when we fished the Madison in 2012.
Here's a photo of Gary Borger chasing the trout. He had gel spun on his reel so he had 100 yards of backing and he needed it. The gel spun is so thin that it adds very little drag when cutting through the water.
Gary landed the fish 700 yards downstream from where he hooked it well below the green roofed home you see on the far bank. In this photo he is already over 100 yards below where he hooked the fish and the fish is well into the backing.
Here's where Gary hooked the fish, in the slick behind the submerged boulder. Notice that the rod tip is bent and the line is tight. I happened to take the photo just as he hooked the fish in his story.
He writes about it here:
Gary Borger » Blog Archive » Runaway Trout
Of the three fish I hooked, I know that one was fair hooked and the other two may have been foul hooked. Those are the only trout that have taken me into backing in almost 30 years of fly fishing.