Thankfully after about 30 minutes of my flatmate and I getting really pissed off (it was like in the same room loud, not just a minor annoyance) I remembered that I had an old car FM transmitter for an MP3 player, after borrowing the batteries from the TV remote, cycling through the stations I was overjoyed to discover that one of the stations was close enough to the radio frequency to blot out the station it was tuned to. The reason I think they weren't home is I turned it on and off a few times but they didn't attempt to turn the radio to another channel or anything like that. I went to sleep with it on.
Technically the FM transmitter was legal at that point but using it to deliberately block someone else's radio is almost definitely illegal.
So my friend bought and built an FM transmitter kit. He put it in his apartment in the closest spot to the guy's stereo. And then he broadcast beautiful silence on that radio station's frequency. The neighbor tried other stations for a while, but it just wasn't the same.
I wonder if a Bluetooth version of that is possible.