Bradbury was born in Coventry, where his father, Bert, was a painter and decorator for the council, and his mother, Joan, worked first for GEC (she was a shop steward) and then at Walsgrave Hospital, where she helped with care in the maternity ward.
Back in Coventry, he became part of a small group of former art students and music fans who included Jerry Dammers, with whom he shared a house.
It was while he was working at the Virgin record store in Coventry that Brad first met Terry Hall, another music fan with similar taste.
Bradbury was asked to join the Specials after their original percussionist ("a strict reggae drummer" called Silverton Hutchinson, according to bassist Horace Panter) had left.
Bradbury played a crucial role in the history of the Specials, the still massively successful multiracial band that first shook up the British music scene in the late 1970s and early '80s by mixing Jamaican ska with punk energy, a bravely political stance and good humour.
With the Specials he toured Britain, packing out dancehalls where followers showed their respect by wearing the uniform of pork pie hats and black-and-white sweaters.
For Dammers, the group’s founder and keyboard player, Bradbury was "highly intelligent, had a mischievous sense of humour which could be very funny, and he played the drums with an incredible amount of energy which was a very important part of the Specials’ sound and live shows".
He looked after his son, Elliot, while his wife, Emily, worked in the fashion industry, and he was involved in a series of projects that included computer programming and maintenance.
The band proved that they could still attract large and enthusiastic crowds, even without Dammers, and set off on extensive tours across the UK, Europe, North America and the Far East.
The live album The Specials: More or Less (2012) was a well-produced reminder that the band could still do justice to their classic repertoire – and that Bradbury was still in powerful form behind the drums.