Why can't helicopters fly at high altitudes?
Don't believe everything you see in the movies. Doubly so if the movie originated in Hollywood.

The world altitude record for a helicopter is 40,814ft (12440m), some 11,000 feet (3353m) higher then Everest, set in 1972 Reference: Wikipedia

In 2005 Didier DelSalle famously touched down on the summit of Everest in an AS350 Squirrel Reference: Wikipedia. See below for a video.

Both feats involved specially prepared aircraft and highly experienced crew.


                
            

As has already been pointed out, a helicopter, just like any other aircraft, has a ceiling above which they cannot fly. Since it's clear that it is possible to fly helicopters to the required altitude in certain conditions, this is just a red herring.

The real reason helicopters don't routinely fly to those altitudes is that they're not designed to. As with everything in aviation, performance is a trade off between engineering, aerodynamics and commercial pressures. 

There's a big market for helicopters that can fly up to, say, 12-15,000ft (3600-4500m). The market for helicopters capable of rescuing climbers from the top of Everest is, at best, limited, and the technical demands of building a craft routinely capable of such a flight will push the costs up dramatically.

In short, they don't because nobody will pay for it.