Bright received his bachelor's degree in petroleum engineering from Texas A&M University in 1943. After graduating, he enlisted in the U.S. Army. He was assigned to the Army Corps of Engineers and deployed to Europe during the Second World War. Bright remained in the Army until 1946, obtaining the rank of captain.
Bright returned to Texas in 1946 and began investing in oil and natural gas leases. Later he expanded to trucking, banks, real estate, and savings and loans. His investments made him a millionaire by the age of 31. In 1990, he was listed as one of the 100 richest people in Texas.
In 1984, Bright (along with 11 limited business partners, including Craig Hall) purchased the Dallas Cowboys NFL franchise from Clint Murchison, Jr. for $85 million.
The Cowboys made the playoffs in his first season as owner, losing to the Los Angeles Rams in the first round and finishing with a 10-6 record. The team's performance declined over the next three seasons, posting records of 7-9, 7-8, and 3-13.
After suffering financial losses, including an estimated $29 million in the 1988 collapse of First Republic Bank Corporation, Bright sold the franchise to Jerry Jones in 1989 for $140 million. Jones made many changes to the team, including firing longtime head coach Tom Landry. Later, Bright claimed that he wanted to fire Landry himself as early as 1987, but general manager Tex Schramm claimed he didn't have a successor in place yet.
An outspoken conservative, Bright was highly critical of John F. Kennedy. In response to President Kennedy's visit to Texas in 1963, Bright co-sponsored a political advertisement in the Dallas Morning News to criticize the President's policies. The full-page ad ran on November 22, 1963, the day that Kennedy was assassinated in Dallas.
He also disapproved of the extent of federal welfare programs. In a 1990 interview with Texas Monthly, he stated, "Our unrestrained welfare programs encourage people to be non-productive and, therefore, hungry and helpless."
Bright was known for publicly supporting his alma mater, Texas A&M University, and its athletics programs. After the Texas A&M football team upset the University of Texas in 1967, he had one million matchboxes with the inscription "Tee Hee Hee: Texas A&M 10, Texas 7" printed and distributed throughout the state of Texas.
He served as the chairman of the Texas A&M University System Board of Regents from 1981 to 1985. In 1991, Texas A&M named him a Distinguished Alumnus. A&M also presented him with the Sterling C. Evans medal in 2002.
Bright donated an unrestricted contribution of $25 million to Texas A&M in 1997. In 2000, he donated another $5 million to support renovation of the school's football stadium, Kyle Field. Two buildings on Texas A&M's campus bear his name: the Bright Athletic Complex and the H. R. "Bum" Bright Building.