[1][2] He lived in North Reading, Massachusetts, for the first two years of his life, after which his family relocated to Vero Beach, Florida, due to employment.

[2][3][4] As a youth, Cobb served as a batboy for the Los Angeles Dodgers at Holman Stadium in Vero Beach for three years of spring training.

While there, he was an all-state pitcher as a junior as he had a 8–2 win–loss record with an 0.62 earned run average (ERA) and 139 strikeouts in 90 innings pitched.

[19] He ended the season with a 11–9 record and a 4.03 ERA in 23 starts covering 136.1 innings at the major league level.

[21] Cobb was struck in the head by a line drive off the bat of Kansas City Royals first baseman Eric Hosmer on June 15, 2013.

Cobb was carried off the field on a stretcher, and transported to Bayfront Medical Center in St. Petersburg, Florida.

It was reported that Cobb suffered a mild concussion and had a cut on his right ear, while all other scans and tests came back normal, and he would be discharged the next day.

[22] Cobb suffered nausea, headaches, and vertigo, and missed two months, making his return on August 15.

[citation needed] In 2014, Cobb endorsed a product designed to help protect young ballplayers from similar injuries: the isoBLOX padded cap insert.

[1] To begin the 2015 season, Cobb was placed on the 15-day disabled list due to right forearm tendinitis, missing what would have been his first Opening Day start.

Due to the health of his arm and his rehabilitation program, Cobb was not allowed to throw his split-finger fastball, his most dominant and often-used pitch.

[32][33] After an injury to Matt Andriese and poor play by Blake Snell coupled with Cobb's flashes of high potential, the Rays stated that they were not actively shopping Cobb, instead planning on utilizing him as a key piece for the regular season and playoffs.

[17] He was named the winner of the Paul C. Smith Champion Award, which goes to the Rays player who best exemplifies the spirit of true professionalism on and off the field.

On June 11, it was revealed Cobb needed to undergo hip surgery, and he was ruled out for the rest of the season.

[41] Batters hit hard-hit balls against him 48.2% of the time, putting him in the worst 4% of baseball pitchers in that category.

He also was in the top 94% among qualified pitchers in "barrel rate," as he gave up a barrel (high exit velocity and ideal launch angle) on only 4.2% of batted balls against him, and in the top 93% on his career-best "chase rate" (the percentage of pitches outside the strike zone that he induces batter to swing at).

Spencer Steer broke up the bid with two outs with an RBI double (scoring a runner who had walked earlier in the inning).

[53] Cobb made 28 starts for San Francisco in 2023, registering a 7–7 record and 3.87 ERA with 131 strikeouts across 151+1⁄3 innings pitched.

Following the season on October 30, it was announced that Cobb would undergo left hip surgery to address his labrum and ongoing impingements.

[55] While beginning the 2024 season rehabilitating from surgery, Cobb began to experience shoulder inflammation in his right arm, and was transferred to the 60-day injured list on April 20, 2024.

[64] Cobb's brother, R.J., is a United States Army officer who served in Iraq and Afghanistan and was awarded a Purple Heart.

Their mother, a nurse practitioner, died in December 2005 at 49 years of age as the result of a stroke, when he was a senior in high school.