Aldredge was born in Dayton, Ohio, the son of Lucienne Juliet (née Marcillat) and W. J. Aldredge, a colonel in the United States Air Force. His most notable roles have been on the stage. He has been nominated for five Tony Awards for his roles in Sticks and Bones (1972), Where's Charley? (1975), The Little Foxes (1981), Passion (1994) and Twentieth Century (2004). He also starred in the original Broadway productions of Vieux Carré (1977), On Golden Pond (1979), and Into the Woods (1987) and the 2000 off-Broadway revival of The Time of the Cuckoo.
Aldredge's film credits include What About Bob (1991), Cold Mountain (2003), and The Assassination of Jesse James by the Coward Robert Ford (2007). He appeared as Dr. Robinson in Disney's The Adventures of Huck Finn (1993).
Aldredge also played the recurring role of Hugh DeAngelis, the father of Carmela Soprano, on The Sopranos from 2000 to 2007. Since 2007, he has played the recurring role of Uncle Pete, alongside Glenn Close, in Damages