Analysis of information sources in references of the Wikipedia article "Walter Brennan" in English language version.
...both wins said to be due to the high level of support he received from the extras union whose members were allowed to vote in the Academy Awards from 1936 to1940. His third win on his third nomination for 1940's The Westerner caused such a scandal that the extras' voting rights were taken away.
In the early years of the Academy Awards, extras were given the right to vote. Brennan was extremely popular with the Union of Film Extras, and since their numbers were overwhelming, he won each time he was nominated. Some say that his third win led to the disenfranchisement of the Extras Union from Oscar voting.
While there, he suffered an injury to his vocal chords from exposure to mustard gas that left him with his screen trademark: a distinctively reedy, high-pitched voice that became a favorite for celebrity impersonators for decades.