If you're intentionally carp fishing you should have like 2x tippet on, and be able to put some serious pressure on. But yeah, if you're fishing big water for other stuff and happen to hook a carp, I can certainly see it.
Big river trout and current, like Afish's story, yep, they'll get you there, though you have little hope of recovering in heavy current like that.
Ocean, oh heck yes. I was on a charter once, not fly line but regular line, and we hooked a blue marlin which spooled us. We're talking big equipment and line which must be like 100 lb test. And we hooked him just off the teaser just feet from the boat. They had another rod ready for just that occasion! Thick line, steel clip on the end. Hook up the first rod and throw it overboard. When you recover enough line back, grab the first rod and fight it on there again. We caught and released the fish, though it was not "my" catch. 6 guys on the boat taking turns on dolphin and tuna. It was unfortunately not my turn when the marlin came up.