Cassado was a Spanish contemporary of Pablo Casals, who lived from 1897 to 1966. In fact, he may have been Casal's youngest pupil, when he studied with him in Paris in 1910. He was born in Barcelona, the son of a church musician. His father started teaching him music when he was five, and at the age of seven he began cello lessons with a prominent Barcelona cellist, who worked at the Mercedes Chapel with his father.

When Cassado was nine years old, he played his first public performance, where he was heard by Casals, who immediately offered to give him lessons. He was given a scholarship by the city of Barcelona to go to Paris and study with Casals there. He also studied composition with Manuel de Falla and Maurice Ravel.

At the end of World War I, Cassado started touring internationally, and became a world famous cellist. He played under most of the leading conductors of his time, including such greats as Furtwangler, Beecham and Weingartner. His performance of the Brahms Double Concerto with Joseph Szigeti was especially appreciated.

Cassado loved Italy, and settled in Florence, where he lived for over thirty years. As a cellist he was more austere and noble, than flamboyant in his approach. He was a good composer, and his pieces are still played today, in particular his Requiebros, and his Concerto in D Minor, which he dedicated to Casals.

In 1964 Cassado premiered six unpublished cello sonatas of Boccherini, and performed them on a Strad cello that was once owned by the composer. Eve Barsham, his accompanist, had discovered the manuscripts in the archives of the Duke of Hamilton in Scotland. Cassado died in 1966 of a heart attack, after a strenuous tour of a flood stricken area of Florence where he was raising funds for those who had been devastated by the natural catastrophe.