Yes, the majority of stars in film, music, and television usually keep theawards that they win.
However, in some cases some stars have refused to accept awards out of protest for a certain cause they believed in. An example would be at the 1972 Academy Awards, when Marlon Brando won for, "Best Actor, for the Godfather." He didn't attend the ceremony, but, sent an American Female Indian, to read a speech refusing the award, because of the way the movie industry had portrayed and disciminated against the American Indians.
An example in music world be in at the 1980 Country Music Association Awards. When the soundtrack from the hit movie, "Coal Miner's Daughter (the story of the life and times of country great Loretta Lynn)," won the award for, "Album of the Year." Owen Bradley, who produced the LP accepted the award, but, announced the he would give the award to actress Sissy Spacek, for she deserved it. Sissy Spacek also won the Oscar for, "Best Actress of the Year," for playing the role of Loretta Lynn in the movie.