Originally playing its home games at Dyckman Oval, in 1940 the club moved to the Polo Grounds. In constant competition with the New York Black Yankees for the affection of New York fans, the Cubans consistently fell short, despite fielding stronger teams.
In 1935 the Cubans earned the right to play the Pittsburgh Crawfords for the Negro National League crown but lost the 7-game playoff series. In that year the Cubans pitching staff was led by Hall-Of-Famer Martin Dihigo, a Cuban star regarded by many experts as the greatest player of all time.
Coming up short again in 1939 the Cubans lost the NNL title to the Homestead Grays. Although owner Alex Pompez (a prominent Harlem underworld figure) consistently fielded strong teams, the Cubans did not capture a league title until 1947 when they took both the pennant and the Negro World Series championship defeating the Cleveland Buckeyes. The 1947 squad was led by future major league star “Minnie” Minoso and Luis Tiant, father of the Boston Red Sox great.