Take an opponent who is immune to conventional damage as we understand it. Superman-esque, but without the actual super strength, or some D&D monster with lots of DR, or what-have-you.
Grapple him in such a way that he's left with 2 choices -- accept being grappled and going where you want him to, or resisting and breaking his arm / leg / wrist / neck. (For those not sure what I mean, there's a nifty hold in Aikido that results in, if you don't relent, losing your wrist, elbow, and possibly shoulder all at once...)
The guy doesn't relent. So what happens? Does he suddenly have an explosion of breaking bones? Does he find he can not move because the bones / tendons / muscles can't tear?
On a related note...
Take the same guy. Strangle him. Does he actually strangle if he can not be harmed by conventional (let's say 'non-magical') damage? What happens?
Just some food for thought, I'm curious what people have to offer, and if it will alter my views.
(Which are -- you can cause someone who is 'nigh invulnerable' or 'invulnerable' to harm themselves under most circumstances, and unless someone is explicitly immune to suffocation, you'll be able to suffocate them)