It's extremely possible depending on the frequency, the altitude of the aircraft and the distance to the transmitter you are receiving. It's called multipath interference, and is due to your receiving one signal directly from the transmitter and a second signal reflected by the airplane. If the two signals reach your antenna in phase, the strength of the two signals will add, but if they reach your antenna out of phase, the strength of the two signals will subtract, and reception will be weaker. As the airplane flies by, you'll hear alternating periods of stronger and weaker signals.
When I used to watch television using an antenna, rather than via cable, I could watch the picture fade in and out when an airplane flew over. If you listen to FM in your car, you might well hear a similar effect caused by multipath caused by reflection from buildings and other objects as you drive along.
Depending on what model radio you have, there could be image issues. This is particulary true of dual conversion scanners as opposed to the higher-end triple conversion models.
I used to consistently hear air traffic control on 156.800. I found out that I was hearing a low altitude ATC sector just west of me and the real frequency was 135.400. These days, I often get TV signals on the rail band. Not sure whether it's cable leakage or the TV IF oscillators/amps.