In most aircraft and certainly in jets, you have six primary instruments: airspeed indicator, attitude indicator (artificial horizon), altimeter, turn indicator, heading indicator, and a horizontal speed indicator.
The attitude, turn, and heading indicators rely on gyroscopes that are propelled by a vacuum pump and/or electric motors.
Airspeed, altimeter, and vertical speed indicators rely on a pitot-static system. The pitot tube must be exposed to the air that is uninterrupted by the plane's passage. The static port(s) must be positioned where the air is calm and undisturbed.
Each instrument of the pitot-static system relies on pressure differentials, but only the airspeed indicator is reliant on both the pitot tube and the static port. The other two instruments rely on the static port and vents or calibrated leaks.
Part of a pilot's primary training and certainly part of instrument training is in recognizing failures in this crucial system. For example, let's say you take off and your airspeed slowly drops to zero but you remain flying. There's a good chance you've got a blocked pitot tube. This is a simple example, but pilots are trained on this and other failure scenarios during primary and instrument courses. Pilots that want to die of old age train themselves beyond what's required by these courses.
At lower altitude in visual flying conditions, these failures aren't as deadly, primarily because one can still see the horizon. But high altitudes remove most of the visual cues, requiring pilots to trust the hell out of those instruments...except when they can't.
Someone else said it, but these guys lost situational awareness; I'm guessing that all of the warnings going off just overloaded them and caused them to overlook the clues that would have saved them. I've personally never flown anything bigger than a six-place Beechcraft, but I've had enough scary moments with buzzers and lights flashing to appreciate just how much pressure that situation would have caused.