Higginbotham received a Bachelor of Arts degree from the University of Alabama in 1960, attending on a tennis scholarship offered to him by then-Athletic-Director Paul "Bear" Bryant and serving as the team captain.

Higginbotham was nominated by President Ronald Reagan on July 1, 1982, to a seat on the United States Court of Appeals for the Fifth Circuit vacated by Judge Reynaldo Guerra Garza.

[4] In 1986, when the nomination of Robert Bork to the Supreme Court of the United States was flailing, Higginbotham was widely considered the leading replacement candidate.

For many years, Higginbotham was a faculty member at the Federal Judicial Center and, as an appointee of Chief Justice William Rehnquist, the chairman of the Advisory Committee on Civil Rules.

He has been a leading proponent and former chairman of The Center for American and International Law, a Dallas-based organization which aims to train foreign and domestic lawyers and police officers, a Fellow of the American Bar Association, chairman of its Appellate Judges Conference, member of the Board of Editors of the ABA Journal, and advisor to the National Center for State Courts on its study of habeas corpus.