Born in Port Arthur, Texas,[2] Fleckman credits Byron Nelson, Carl Lohren, and Jim Hardy with teaching him how to play golf.

In 1965, he won the NCAA Championship while at the University of Houston,[2] where he was a three-time All-American member of the golf team: third-team in 1964, first-team in 1965 and 1966.

Fleckman finished in a tie for 18th place and was the low amateur, a stroke ahead of Bob Murphy, who shot 69 in the final round.

[2] At Oakbourne Country Club in Lafayette, Louisiana, he sank a 30-foot (9 m) birdie putt on the first extra hole to defeat Jack Montgomery and take the winner's share of $5,000.

[11] Fleckman is only one of five other players to win his first tour event as a professional,[4] and has since been joined by Ben Crenshaw (1973), Robert Gamez (1990), Garrett Willis (2001), and Russell Henley (2013).