Bolsinger graduated from McKinney North High School in McKinney, Texas, where he was a three-year letterman in Baseball. As a senior, he pitched to a 9–3 win–loss record with a 1.66 earned run average (ERA) and 116 strikeouts and took his team to the state semi-finals, beating the Highland Park High School team that included future teammate Clayton Kershaw. He was drafted by the Cleveland Indians in the 34th round of the 2006 Major League Baseball draft and his high school coach said "Obviously he has the talent to play at the next level. It's not much of a surprise to me that he got drafted." He chose not to sign with the Indians because of the low round and, even though he had signed a letter of intent to Texas Tech University, he chose instead to attend Grayson County Community College in the hopes of being drafted again the next year. In 10 appearances for Grayson, he was 6–1 with a 2.93 ERA and struck out 48 batters.

Bolsinger played in the Texas Collegiate League during the summer of 2007 and chose to transfer to the University of Arkansas for the 2008 season. As a sophomore in 2008, he was 4–1 with a 3.73 ERA in 15 games, most of them out of the bullpen. He missed the start of the 2009 season as he was battling a bad case of mononucleosis He was 6–4 with a 2.99 ERA in 30 games pitched that season as the Razorbacks advanced to the College World Series, where they finished third. He was drafted in the 33rd round of the 2009 Major League Baseball draft by the Oakland Athletics, but chose to return to Arkansas for his senior season. Bolsinger began the 2010 season as the teams closer and earned a spot on the National Collegiate Baseball Writers Association "Stopper of the Year Watch List." However, he made a successful transition to the starting rotation during the season and earned Southeastern Conference pitcher of the week honors on May 10 when pitched eight innings and struck out 11 while allowing only two hits against the University of Mississippi. For the season, Bolsinger appeared in 23 games (13 as a starter) and was 6–5 with a 4.81 ERA.

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