Barnet was born in New York City. His parents divorced when he was two, and he was raised by his mother and her grandparents. His grandfather was Charles Frederick Daly, a vice-president for the New York Central Railroad, banker, and businessman.
Barnet attended various boarding schools, both in the New York and Chicago areas. He learned to play piano and saxophone as a child. He often left school to listen to music and to try to gain work as a musician.
He was one of the first bandleaders to integrate his band; the year is variously given as 1935 or 1937. He was an outspoken admirer of Count Basie and Duke Ellington; Basie once lent Barnet his charts after Barnet's had been destroyed in a fire. Throughout his career he was an opponent of syrupy arrangements.
Barnet was at the height of his popularity between 1939 and 1941, a period that began with his hit "Cherokee." In 1944 he had another big hit with "Skyliner". In 1947 he started to switch from swing to bop. During his swing period his band included Buddy DeFranco, Roy Eldridge, Neal Hefti, Lena Horne, Barney Kessel, Dodo Marmorosa, Oscar Pettiford, and Art House, while later versions of the band included Maynard Ferguson, Doc Severinsen, and Clark Terry.
In 1949 he retired, apparently because he had lost interest in music; he was able to retire when he chose because he was one of the few heirs in a very wealthy family. He occasionally returned from retirement for brief tours but never returned to music full time.