Rules and regulations?

I just returned from Disneyland, the happiest place on earth. On every single ride they tell you things like, "Keep your hands, arms, feet, inside the ride at all times.", "secure belongings", etc., but what was on EVERY ride even when either one of those weren't, was "Watch your children."
I'm assuming that they tell you these precautions so if anyone disobeys them and gets hurt, they can't sue for injuries because they did tell you not to do it. So, if they tell you to watch your children, and the child does anything that hurts them, can Disney say that they aren't responsible because someone was supposed to be watching them?
So that covers all the supervised children, but what about all the other people who aren't supervised? Are they expected to use common sense and listen to the rules? I've heard the story about the guy on the Matterhorn and when I went on that ride they were very meticulous about checking seatbelts and that they were secured.
So what does happen when someone gets hurt on a Disney ride anyways?