O'Brien was the third son of musician Haskell "Hack" O'Brien, a well-known drummer for several big band era ensembles. When he was an infant, his mother thought he resembled a bear cub, and tagged him with his lifelong nickname. He has two older brothers, Haskell O'Brien, Jr., a trumpet player, and Warren O'Brien, also a drummer.
Although born in Burbank, O'Brien grew up in the Los Angeles suburb of Shadow Hills, California between Sun Valley and Sunland. He began studying music at the Carl Babcock School of Music from age five, and later performed with the Roger Babcock Dixieland Band at charity events and for local television. He would eventually graduate in 1962 from Hollywood Professional School, where he was president of his class.
O'Brien, like Annette Funicello, was personally selected to audition for The Mickey Mouse Club by Walt Disney, in the spring of 1955. Disney had been alerted to him by a staff member, who caught his live performance at a charity gala.
Though he had little prior experience in singing or dancing, O'Brien was placed on The Mickey Mouse Club's first-string "Red Team" right from the start. He quickly picked up enough dance skills to perform in musical numbers, though his solo performances remained centered around his drums. He remained with the show for all three seasons (1955–1958) of original programming, and after filming stopped, went on live-performance tours with other Mouseketeers to Australia in 1959 and 1960.
Following Disney, he joined the Lawrence Welk organization for two years, having performed with the "Little Band", doing guest bits on the maestro's show, as well as other television series. After graduating from high school, Cubby started touring with Spike Jones, playing show tunes and dance music with the formerly manic bandleader's final group.
O'Brien was a contestant on the ABC game show The Big Showdown in the mid-1970s, winning $5,000 for rolling "Show Down" during the timed dice roll round.
In 1980, O'Brien reunited with his fellow Mouseketeers for a television special, in which he sang and danced, and, of course, played drums. He also joined some of these same Mouseketeers for live shows on fall weekends at Disneyland during the early 1980s.
Since the eighties, O'Brien's career has predominated around performing for Broadway productions, as well as for live shows by performers like Bernadette Peters. He based himself out of New York City instead of the West Coast for many years, though he has since moved back.
In 1978, O'Brien married Terry Wilemon, whom he met in Las Vegas while touring with the Carpenters.