Hibbert was born on Long Island, New York, the son of actor Geoffrey Hibbert. He has one sister. He was brought up in Britain, where he attended the Royal Academy of Dramatic Art. He returned to the US in the mid-1980s.
Hibbert had a starring role as Faulconbridge in the BBC's production of The Life and Death of King John, published in 1984. He has appeared on Broadway and in major regional theatre productions, worked in television as a series regular and guest star and also had roles in major films. In 1993 he won an Obie Award for his co-starring role of "Sterling" in Paul Rudnick's Jeffrey. His "Frederick Fellows/Philip Brent" in the National Theatre revival of Noises Off (presented at the Brooks Atkinson Theatre) was called "delightfully discombobulated" by one reviewer. Hibbert was in the Broadway musicals The Drowsy Chaperone and the 2007 premiere of Curtains (which reunited him with his Frasier co-star David Hyde Pierce). He appeared on Broadway as "Mr. Praed" (the architect) in Roundabout Theatre's 2010 production of Mrs. Warren's Profession starring Cherry Jones. He appeared on Broadway in the new musical It Shoulda Been You in 2015, again working with David Hyde Pierce, this time with Pierce as the Director.
