Midlifeflyer, the speed is the key factor here, however its closely linked with altitude. At low altitude you can fly 340 KIAS and still only show around mach .5. In most commercial jets you are then well below your critical mach number. but at 41000 340 KIAS would be around mach 1.2
In a jetfighter you probably have a MCRIT just below mach 1, while as a B737 maybe has a MCRIT just above mach .7 (dont quote me on that). MCRIT is defined as your free stream mach number at which you first will start to see supersonic airflow over some part of the aircraft, normally the upper sides of the wings since that is where the difference in speed of the free stream airflow and the airflow disturbed by the aircraft is the greatest.
When you get supersonic airflow over the wings a shockwave is created. It acts like a wall and either changes the laminar boundary layer to a turbulent one or simply destroys the boundary layer thereby reducing the coefficient of lift. If speed is increased in an aircraft not designed for high transonic to supersonic speeds, the boundary layer will eventually separate causing a high speed stall.
A C172 that cruises at around Mach .15 wont have any compressibility problems. But since its wing is designed for low speed flight it would get compressibility problems long before a commercial jet.