Thompson was born on December 11, 1948, in Baltimore, Maryland. He has an older brother, who played in the drum corps. At elementary school, he learned to play the flute and read music. At eleven, Thompson took up the drums, receiving lessons from James Harrison, a professional jazz drummer from whom he learned his rudiments. Thompson practiced by playing along with albums by jazz musicians Miles Davis, Max Roach and Art Blakey. From there, he moved on to studying records by drummer Elvin Jones, whom Thompson cites as a major musical influence along with Tony Williams. While attending high school, he studied privately with drummer and percussionist Tony Ames of the National Symphony Orchestra for one semester. Thompson's practice focused on mastering drumming rudiments using a book published by the National Association of Rudimental Drummers. He started to play his first live gigs two years later in local venues. Still underage, Thompson went as far as to draw a moustache on his upper lip using an eyebrow pencil "because the club owners were worried about me playing there". He played as many as three jam sessions a week.
Among his first major jobs was a short tour, mainly across Canada, with singer Ben E. King. In 1970, he played with organist Jack McDuff, followed by gigs with other local groups, including time in Boston with keyboardist Webster Lewis. In 1971, Thompson returned to Baltimore and studied music at the Community College of Baltimore County for two years, where he learned the flute and coached a basketball team at the Rec Center. He played as part of the house band in a club that supported visiting soul artists. Thompson built a reputation as a session drummer. One of his first bands was Doc "Soul Stirrer" Young and the We Four Trio. Since the early 1970s, Thompson has played with the jazz rock band Air Pocket, a band with the Fowler Brothers at its core.
In 1973, shortly before Thompson was to start a four-year course at another music school, he had auditioned to join Frank Zappa and The Mothers of Invention and won the position. He was a friend of their tour manager, Marty Perellis, also of Baltimore, and landed an audition in Los Angeles after learning of Zappa's wish to use two drummers in his group. Thompson recalled the audition involved him jamming with the band for a solid hour without a break. "We just drifted in and out of so many different kinds of feels and grooves." His time in Zappa's band was challenging because of the leader's "incredibly difficult music" which involved as much as 40 hours of weekly practice for four to six weeks before a tour. Thompson played on several Zappa albums, including Roxy & Elsewhere (1974), One Size Fits All (1975), Studio Tan (1978), Sleep Dirt (1978), and You Can't Do That on Stage Anymore, Vol. 2 (1988).

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