Bamji was born in Bombay, India, to a Parsi family that was residing in Bahrain. His father, Esadvaster, was the regional representative for Norwich Union Life Insurance Society. His mother, Roshan, was a homemaker, and both his parents were active in various local charities.
During his last couple of years in undergraduate school he began acting at Columbia's first professional theatre, Trustus, where the iconoclastic artistic director, Jim Thigpen took him under his wing. At Trustus he played a variety of roles, including Pale in Burn This, Torch in Beirut, Danny in Danny and the Deep Blue Sea, Peter Patrone in The Heidi Chronicles and all the parts in Eric Bogosian's solo play, Drinking in America. After studying for an MFA in Theatre at USC, Bamji moved to Washington D.C. to finish his degree as an apprentice at The Shakespeare Theatre. In 1994 he was cast in Eric Bogosian's SubUrbia at Lincoln Center's Mitzi Newhouse Theater, and he and his then wife, Erin Thigpen, sold the car and moved to New York City. Bamji has appeared on stages in New York, Chicago, San Francisco, Washington D.C. and Los Angeles and major regional theaters around the United States. He has played leading roles in world and American premieres of plays by playwrights such as Tom Stoppard, Tony Kushner, Naomi Wallace, Rebecca Gilman and Eric Bogosian. In 2007 he was invited by director Simon McBurney to co-write and perform in a new play with the British company Complicite. The piece was to revolve around the relationship between two pure mathematicians who lived at the turn of the 20th century, the self-taught genius, Srinivasa Ramanujan and Cambridge University don, G. H. Hardy. Bamji had been interested in this story for a few years and was working on a film script when he was approached by McBurney. The result was A Disappearing Number, which won the Laurence Olivier Award and the Critics Circle Theatre Award for Best New Play and the Evening Standard Award for Best Play. Over the next four years, A Disappearing Number toured Europe, Australia, India and the United States and finished its universally acclaimed run at the Novello Theatre in London’s West End. Bamji's television credits include Law & Order and Law & Order SVU and his film credits include The Sixth Sense, Unbreakable, Analyze That, Ashes, Justice and The War Within, for which he received an Independent Spirit Award nomination. In 2015 he received an Obie Award for his performance in Roundabout Theatre’s production of Tom Stoppard's Indian Ink. He has narrated more than twenty audio books, including The Gambler by Fyodor Dostoevsky, Siddhartha by Herman Hesse, Camille by Alexandre Dumas, The Hungry Tide by Amitav Ghosh, and The Enchantress of Florence by Salman Rushdie, for which he received an Audie Award nomination.
