Anne Bancroft is a classic example. Her real name was Anna Maria Italiano, and she was told outright that it was, well, too Italiano, and she'd never have a successful acting career with that name. So she changed it to Anne Marno (Marno being a combination of her middle and last names).
Nope, they said. Still too “ethnic-sounding”. So she chose “Bancroft”, because, she said, she thought it sounded “dignified”. (I bet Mel Brooks would still have married her no matter what her name was, though.)
The same story happened over and over; non-Anglo people took Anglo stage names in the hope of it increasing their chances of Hollywood success.
So Issur Danielovitch became Kirk Douglas.
Volodimyr Palahniuk became Jack Palance.
Tadeusz Konopka became Ted Knight.
Charles Bunchinsky became Charles Bronson.
Jacob Julius Garfinkle became John Garfield.
Margarita Carmen Cansino became Rita Hayworth.
Alphonso D'Abruzzo became Alan Alda.
Emanuel Goldenberg became Edward G. Robinson.
Frederick Austerlitz became Fred Astaire.
Dino Crocetti became Dean Martin.
Bernard Schwartz became Tony Curtis.