Born into London's East End Jewish community, Crombie was a self-taught musician who began playing the drums at the age of fourteen. He was one of a group of young men from the East End of London who ultimately formed the co-operative Club Eleven, bringing modern jazz to Britain. Having gone to New York with his friend Ronnie Scott in 1947, witnessing the playing of Charlie Parker and Dizzy Gillespie, he and like-minded musicians such as Johnny Dankworth, and Scott and Denis Rose, brought be-bop to the UK. This group of musicians were the ones called upon if and when modern jazz gigs were available.