You can hear it because your headphone wires will pick up wavelengths about the same length (ever heard of people making AM radios with wire and a tin can?). My understanding and I hope I can be corrected if wrong is that this only will work for FM transmissions.
I've heard the best thing to do in these situations is to politely inform your neighbor of the situation and to investigate a solution. Probably some filter on his transmitter would do the trick. But I'm sure if he's serious about radio, he'll have a grand time fixing it...
If a non-radio device is picking up radio signals it can never be the Transmitter which is at fault.
It must be RF breakthrough caused by bad (eg cheap) design in the device.
Source: Retired EMC engineer.
P.S. I've never heard of people building an AM radio with a wire and a tin can. And if it were possible, any fault would be with the poor receiver not the transmitter.
(Capacitor can be made out of foil and paper, crystal out of the graphite from a pencil and a razor blade. The radio pioneers were pretty inventive. Google early crystal radio receivers.)
"..any fault would be with the poor receiver .." is correct, but if you took the time to build your own crystal set, definitely go see the ham next door, he most likely has the parts in his component stash to add some better RF selection/rejection to your radio.
Just like most of you like to tinker with code, we like to tinker with radios.