Bolet was born in Havana in Cuba and studied at the Curtis Institute of Music in Philadelphia, where he himself taught from 1939 to 1942. His teachers included Leopold Godowsky. In 1942 he joined the US Army and was sent to Japan. While there, he conducted the Japanese premiere of The Mikado.
Bolet is particularly well remembered for his performances and recordings of large-scale Romantic music, particularly works by Franz Liszt.