What happens if a helicopter engine fails at 20,000 feet?
Yes. See the Wikipedia's article on autorotation!, which mentions a real-world example of the case you are describing:


  The longest autorotation in history was performed by Jean Boulet in 1972 when he reached a record altitude of 12,440 m (40,814 ft) in an Aérospatiale Lama. Because of a −63 °C (−81.4 °F) temperature at that altitude, as soon as he reduced power the engine flamed out and could not be restarted. By using autorotation he was able to land the aircraft safely.


The biggest concern here is the disk loading of the helicopter's rotor - the greater the disk loading (the mass of the aircraft related to the rotor's area) the worse the autorotation capabilities. See this Google Books link for an in-depth discussion of the autorotation capabilities of different aircraft based on their characteristics.

The weight of the very rotor also plays a significant role here - the heavier the rotor, the more kinetic energy it stores, and the greater the probability of a successful entry into the autorotation state. See the Osprey's design, for example - it has light propellers on each wing which are practically unable to autorotate because of their small mass.

As you can see, the aforementioned helicopter - Aérospatiale Lama - is a light aircraft which is geared towards high altitudes - it has three propeller blades, which translates to low disk loading, and sufficiently heavy blades to allow easy entry and execution of the autorotation stage.