alt Hacker NewsI remember few years ago listening to a radio interviewer with popular folk singer, the interview was over the phone so host at the end said that some day this singer should visit them in studio for proper interview, to which the singer replied: "If I visit you in studio for interview, will you play some of my songs?", to which the host with slight embarrassment admitted that they are not allowed to play outside the playlist.
It depends on the station.
The BBC often does as you say and even if the interview is down a phone line they will play a record. Some shows have the musicians playing live. Other UK stations do similar.