Bunker played in his first band The Warriors in the 1960s. With Mick Abrahams he later formed the band known as McGregor's Engine. In this early career, playing in small venues, Bunker had an extraordinary non-matching drum kit, composed of bits and pieces of various manufacturers’ equipment.
Between 1967 and 1971 he drummed for Jethro Tull, leaving after Tull's release of their most popular album to that date, Aqualung, to get married and spend more time with his new wife. He was replaced by Barriemore Barlow, a school friend of the group's singer/songwriter Ian Anderson. On his decision to leave Jethro Tull at the height of its commercial success, Bunker stated:
"I had always told Ian, 'If I find the right lady, I'll be gone,' and I did just that. That was just at the start of their world touring in '72, and I wouldn't have been back in England for ages at a time. So I thought that I might as well end it then. Besides, Barrie was always in the background anyway, so I knew I wasn't going to put them in a difficult situation. You must understand that back then, we didn't have any time off; it was non-stop work, and I wanted to spend time with my wife."