Brick Joke/Radio

It happens in the third episode of the original Hitch Hikers Guide to The Galaxy series. In order to reduce stress and nervous tension the narrator reveals certain facts early in the episode. Amongst these facts is that a nuclear missile attack will result in a bruised upper arm (amongst other things). To maintain mystery however, to recipient of this bruise is not revealed, "since it is of no significance whatsoever". No mention is made of any bruised arms for the rest of the episode. The Stinger:

Arthur Dent: I'm sorry, but I'd probably be able to cope better, if I hadn't bruised my arm.

On The Jack Benny Program, the Running Gag "Train leaving on track five for Anaheim, Azusa, and Cuc... amonga!" eventually turned into a Brick Joke, with increasing lengths of unrelated dialogue between the "Cuc" and the "amonga!"
In Hamish And Dougal Hogmanay Special, an early gag was the bells (not a typo) freezing off a brass monkey, and subesquently off anything else that had bells. This was then not mentioned for the rest of the episode ... until they waited for the chimes of New Year at the end, and realised that, yes...
When WCBS-FM in New York City became JACK FM in 2005, the last song played was "Summer Wind". When it switched back two years later, they abruptly cut playing "Don't Stop Believin'" and switched to... the tail end of "Summer Wind".
A brick with relatively short air-time: in one episode of Hello Cheeky, Tim announces that they'll be broadcasting Rigoletto in two parts. This is immediately followed by a cry of "Rigo--", and fifteen minutes later, we hear a yell of "--letto!"