Yes, we have cave movies like The Descent, The Cave (2005), and Sanctum. But I was watching The Shawshank Redemption the other day, and there's that scene where Andy is squirming through a tight tunnel while dragging his pack behind him, something I've done underground a time or two. In a cave though, not as part of a prison break!
Speaking of prisons, The Count of Monte Cristo (2002) has some tunnelling, digging, and worming around underground.
The first Rambo movie (First Blood) has a sequence where he's finding his way through an abandoned mine.
First Knight (1995) has Prince Malagant and company in abandoned castle built on top of a cave or mine, with lots of interesting rocks and running water. In fact, when they are trying to find their way out of the underground complex Lancelot says, "if the water can get out, so can we" when selecting a passage. Which is something I've heard off an on over the years if you're ever lost in a cave. Anyway, if that's a real location I'd like to know more about it.
Hmm, just found this on First Knight- A slate mine became the fortress lair of Arthur's enemy, Malagant. Box said: "No more corny castles." He knew there were slate pits in Wales, so he decided to take Malagant underground into his own strange, cavernous world.
Batman Begins has the discovery of the bat cave.
I'm sure there are others.