I remember reading this passage in the "Feynman Lectures", where Dr. Feynman describes an experiment in which a theoretical metal rod of length equal to the distance between Mars and Earth is arranged between mars and earth. Then the "rod handler" on earth gives the rod a push upwards. So the question posed is, will the rod instantaneously move backwards on Mars? He then proceeds to say that actually the movement will be felt roughly after the time it would take for sound to propagate from Earth to Mars i.e. at the speed of sound. This is because the atoms in the rod actually propagate their position at the speed of sound. See also this Phys.SE post and links therein. My question is: If this were true, how is supersonic travel possible? Won't there be a scenario where the atoms in the components of the aircraft won't "catch up" because the propagation of the new positions is slower than the actual speed? Won't the aircraft disintegrate?
Sound traveling through an aircraft propagates at a speed relative to the aircraft, so no, it can't go so fast that the atoms can't communicate via vibrations any more. If sound moves through the aircraft at 1000 m/s and the plane is flying past you at 1500 m/s, you will observe a sound wave to travel from back to front at 2500 m/s.
When an jet accelerates, it does indeed compress a bit, since it takes a short time (hundredths of a second) for the front of the plane to know the back is accelerating. However, the plane feels only acceleration, not velocity. The plane cannot in principle detect its own velocity due to the principle of relativity. If the plane accelerated hard enough, yes it could buckle up on itself.
You may also be interested to know that shocks frequently travel through materials faster than their sound speeds, and that you could in principle send a message along a pole faster than the speed of sound in the pole. It is not a fundamental limit like the speed of light.
Also, when people talk about supersonic travel, they mean the craft is traveling through the air faster than the speed of sound in the air, not that it's traveling through air faster than the speed of sound in its own material, which wouldn't be very physically meaningful, and also never really happens since the speed of sound in metals is very fast.