Bradley was born to a wealthy family in Cleveland, Ohio, the son of M.A. Bradley and grandson of his namesake, Captain Alva Bradley. He graduated Cornell in 1908. He was president of the group that bought the Cleveland Indians in 1927 for $1 million, and which in 1946 sold the team to Bill Veeck. While he was the team's president, he was not the majority shareholder. Other members of the ownership group included his brother, Charles C. Bradley, with whom he invested $175,000, John Sherwin Sr. ($300,000), Percy Morgan ($200,000), Newton D. Baker ($25,000), attorney Joseph C. Hostetler ($25,000) and the Van Sweringen brothers ($250,000). During his tenure the team signed teen age strikeout king Bob Feller in a controversial move that had to ultimately be resolved by baseball commissioner Kenesaw Mountain Landis. During the 1940 season Indian players, led by Mel Harder and Ken Keltner, came to him demanding that he fire team manager Ossie Vitt. The Indians were labeled "Crybabies" for doing so and lost the pennant race on the last day of the season. Bradley went out on a limb by hiring then 25-year old Lou Boudreau as team manager.
Bradley had a number of other business interests. He owned a real estate company and was president and treasurer of the United States Coal Company. He served on the boards of several Van Sweringen companies, the American Shipbuilding Company, Great Lakes Towing Company, and others, and was chairman of Cleveland Builders Supply Company.
He was married to the former Marguerite Andrews and had four children: a son and three daughters.