Although it isn't a name that would have you think Native American, Charles Stevens (1893 - 1964) was one. And an important one to the movies and to the name Bit Actor.
Stevens started in the movies in the 1915 classic, The Birth of a Nation. For the next five years or so, in the silent films, he played a variety of American, Japanese, and Mexican parts. He worked with some of the biggest names in silent film. He made no less than 22 films with Douglas Fairbanks (1883 - 1939), including some of his biggest films, The Mark of Zorro (1920), The Three Musketeers (1921), Robin Hood (1922) and The Thief of Bagdad (1924). He would later make three films with Douglas Fairbanks, Jr. in the 1940s.
In 1930 he plays Injun Joe in the Jackie Coogan version of Tom Sawyer. Although Stevens works regularly in the next decade, he doesn't make many famous movies. Remember, this was the depression and any work was good to have. Plus, the country needed cheap entertainment to keep spirits up, and these movies would help.
In 1939 he plays a drunken Indian in Frontier Marshal. This telling of the gunfight at the OK Corral was almost completely fictional, but it must have inspired My Darling Clementine (1946) which is almost the same story, with similar historical mistakes. Stevens plays the drunken Indian in that one as well. Speaking of Tombstone, Stevens plays an Indian in Tombstone: The Town Too Tough to Die (1942) with Richard Dix (1893 - 1949) as Wyatt Earp.
In 1940 he is in the Tyrone Power remake of The Mark of Zorro. Once again as a Mexican. Stevens is found in quite a variety of westerns, and he played an Indian named Joe it six of them, by my count. He was never a regular sidekick to any popular western star, but he worked with almost everyone at least once. (Except Roy Rogers, for some reason.)
In 1956 he played a slave in The Ten Commandments. I dare you to pick him out!
His final film was The Outsiders (1961). This was a film about an Indian, Ira Hayes, in the Second World War, who helped raise the flag above Iwo Jima. (The second time, for the photo. The real Hayes is the one on the left.) Ira was played by Tony Curtis, and Charles Stevens played Joseph Hayes, in a Bit Part as Ira's father. I haven't seen The Outsiders, but I have in my collection Flags of our Fathers (2006) by Clint Eastwood, that covers the real story of the flag raising.
Stevens has over 220 titles listed on IMDb, and did I mention that Charles Stevens was the grandson of Geronimo, and he was 1/4 Apache?