Bradley attended Madison East High after his family moved from Philadelphia to Madison, Wisconsin, in 1973. He graduated in 1977 and attended Wesleyan University in Connecticut, where he studied Theatre and English Literature, before obtaining a Master's in Theatre at Juilliard.

Months after graduating from Juilliard, Bradley landed his first professional part in Curse of the Starving Class, an off-Broadway production starring Kathy Bates. He appeared on stage in numerous other productions, including Romeo and Juliet at the New York Shakespeare Festival, Three Days of Rain at the Manhattan Theatre Club and Measure for Measure at the Lincoln Center. He also played the title role in Coriolanus at the Folger Shakespeare Theatre in Washington D.C.

It was when Bradley starred in the Broadway production of A Few Good Men that he met The West Wing creator Aaron Sorkin (Sorkin wrote A Few Good Men).

Notable film appearances include Presumed Innocent, Scent of a Woman, A Perfect World, Philadelphia, The Client, Red Corner and Kate and Leopold. Bradley has also guest-starred in hit US TV series The X-Files, NYPD Blue and twice in ER (his first appearance was in the Emmy Award-winning Season 1 episode "Love's Labor Lost").

During his first four years in The West Wing, Bradley has earned three Golden Globe Nominations and three Emmy Nominations in Supporting Actor categories. He won the Emmy for Best Supporting Actor in a Drama Series in 2001.

Bradley currently lives in Los Angeles, California.