Born and raised in Longview in the piney woods of east Texas, Cupit attended the University of Houston.

As a member of the Cougars golf team, he earned All-American honors in 1959 and 1960, turned pro in 1960, and joined the PGA Tour in 1961.

Cupit's best finish in a major was runner-up at the U.S. Open in 1963, when he and Arnold Palmer lost to Julius Boros in a three-way playoff.

After reaching the age of 50 in 1988, Cupit played part-time on the Senior PGA Tour.

[1] PGA Tour playoff record (1–2) Note: Cupit never played in The Open Championship.