Bramlett was an All-State and All-American at Humes High School in Memphis and played college football at Memphis State University (now the University of Memphis), where he was named an honorable mention All-American his senior year. Bramlett signed a professional baseball contract with the St. Louis Cardinals but was kicked out of baseball after a few years after getting into trouble. Bramlett then signed a contract with the Denver Broncos and was named runner-up AFL Rookie of the year behind Joe Namath in 1965.
Bramlett only lasted two seasons with the Broncos, making one appearance in the Pro Bowl before being traded to the Miami Dolphins for a fourth round pick in the 1968 NFL Draft. He played two seasons with Miami, making another Pro Bowl appearance, before being traded to the Patriots along with quarterbackKim Hammond in exchange for Nick Buoniconti in 1969.