Does QM break down at large scales?

They don't "break down"; instead, they approach the classical limit (make the same predictions as classical mechanics). See Bohr's principle of correspondence.
http://en.wikipedia.org/wiki/Correspondence_principle

 

As far as I know I would concur with your opinion. We all have heard sometime in our life that there's the possibility to pass through a wall, even though the probability is so small that the universe would reach the heat death before.

I guess these kind of calculations are made according to the amount of particles in the body that has to pass through such a wall. So I would say that the bigger, the less unlikely to happen.