The four-piece cancelled their debut appearance on the show after the presenter's comments during rehearsals for their live performance of their single 'Love Me Like You'.
According to the BBC, Bacon's introduction went: "What do you get when you put two brothers and two sisters in a band? A big fat melting pot of talent".
The band have been feted as one of the breakthrough acts of 2005, their eponymously titled album is currently at number 12 in the album charts and the band have been nominated for the Mercury Music Prize.
The comment was unfortunate given that the band members are not known for being skinny waifs more associated with boy or girl bands frequently seen on 'TOTP'.
In a statement on the website, the band said: "Due to derogatory, unfunny remarks made during the guest presenter's introduction to our performance, we felt we had to make a stand and leave."
The BBC was left in damage-limitation mode, with a spokeswoman claiming that Bacon was not referring to the band's appearance, but was using a figure of speech to describe their status.
Bacon was said to be "mortified" the BBC by the band's reaction and had tried to apologise to the band, but was denied access by their management.
The unhappy situation adds to the woes of the show, which recently lost more than half of its audience when it was moved from Friday night on BBC One to Sunday night on BBC Two. It picked up just 1.1m viewers on its first Sunday, compared with 2.4m viewers on Friday a month earlier.
The Magic Numbers have previously been good-humoured about their appearance, with lead singer Romeo Stoddart saying in an interview that he looked great when standing next to his bandmates.
Bacon has his own radio drivetime show on 95.8 Capital FM, which a Capital spokeswoman said was unaffected by the fallout from the incident.
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