Tinsley was born in Ruple, Louisiana and raised in Homer, Louisiana in the northern part of the state. He attended Louisiana State University where he played football and baseball and was selected as the captain of both teams. He was considered one of the greatest receivers in college football, earning consensus All-American honors in both 1935 and 1936. As a junior in 1935, Tinsley was named a first-team All-American by the Associated Press, United Press, "Liberty" (selected based on a poll of 1,521 varsity players in all parts of the country), "Collier's Weekly", the Newspaper Editors Association, the International News Service, and "The New York Sun". At the time, at age 19 he was one of the youngest ever named to the All-America team. In 1936, Tinsley repeated as a first-team All-American with each of these publications and also received the first-team designation from the Central Press Association (selected by the captains of more than 50 college football teams at "important universities and colleges throughout the United States"), and the Walter Camp Football Foundation.