The ska group tweeted the news: "It is with deep regret that we say goodbye to our great friend, the world's greatest drummer, our beloved Brad. RIP."
Bradbury joined The Specials in 1979, and continued with the reversioned band The Special AKA, who had a top 10 hit with Free Nelson Mandela.
Bradbury took part in The Specials reunion tour in 2009. He also headed up a band called JB Allstars.
The band's representatives said the drummer died in England but no cause of death was given.
In a statement, his family said: "It is with deepest regret that we have to announce the very sad news that our much loved husband and father John 'Brad' Bradbury passed away on Monday the 28 of December.
"Brad's drumming was the powerhouse behind The Specials and it was seen as a key part to the Two Tone sound. He was much respected in the world of drumming and his style of reggae and ska was seen as genuinely ground-breaking when The Specials first hit the charts in 1979.
"He was an integral part of The Specials reforming in 2008 and toured with them extensively up to the present day. His contribution to the world of music can not be understated and he will much missed by family, friends and fans alike.
"It is the family's sincerest wish that they are allowed the time to remember him privately."
The news comes three months after the band's trombonist, Rico Rodriguez, died.
The band, famed for their 1960s mod-style outfits, had seven UK top 10 singles including Too Much Too Young and Ghost Town.
Founder and songwriter Jerry Dammers dissolved the band in 1981 but they re-grouped and continue to perform and record without their former leading man.
Billy Bragg was one of the first musicans to pay tribute to Bradbury.: "A bad day for good music. First we lose Lemmy, now news that Brad from the Specials has passed away. RIP."
Bradbury was born and brought up in Coventry where the band was formed in 1977.
Music producer Pete Waterman, also from Coventry, expressed his shock at the news of the Bradbury's death.
Speaking to BBC Coventry and Warwickshire, Waterman said: "I always had a good laugh with Brad. He was always proud of being in the band and what we'd and he'd achieved.
"He never left Coventry because he always wanted to be part of the scene... he was tremendous."
John "Brad" Bradbury, drummer with The Specials, has died at the age of 62.