Addy’s debut in acting came at Martha’s Vineyard when he performed in summer theater. He played many roles on the Broadway stage, including several Shakespearean ones, usually opposite actor Maurice Evans. After playing two roles in one of Evans’s productions of Hamlet, he played Horatio opposite Evans’s Hamlet in a 1953 Hallmark Hall of Fame television production of the work, the most prestigious American production of the play seen on television up to that time.

Also on television he played roles on The Edge of Night in the 1950s. He made two guest appearances on Perry Mason: Alton Brent in the 1962 episode, “The Case of the Weary Watchdog”, and murderer Joachim DeVry in the 1966 episode, “The Case of the Tsarina’s Tiara.” Later, during the 1970s-1980s, he played publisher Bill Woodard on Ryan’s Hope and patriarch Cabot Alden on the Agnes Nixon-Douglas Marland serial Loving. His television career also includes guest appearances on The Defenders, The Outer Limits, The Fugitive, Ironside, and The Rockford Files.

In motion pictures, Addy’s career spanned four decades. Robert Aldrich used him as supporting actor in several pictures, such as Kiss Me Deadly, The Big Knife (both 1955), What Ever Happened to Baby Jane? (1962), Hush…Hush, Sweet Charlotte (1964) and The Grissom Gang (1971). In 1976, Addy appeared in Paddy Chayefsky’s Network, directed by Sidney Lumet. They would work together again in The Verdict, in which Addy played a doctor who nearly derails Paul Newman’s case against a hospital for malpractice. Another of Addy’s best-remembered roles was that of Lt. Commander Alvin Kramer, who unsuccessfully tries to warn American officials of the impending attack on Pearl Harbor in Tora! Tora! Tora!.