Cohen grew up in St. Louis, Missouri, in a Jewish family. His parents divorced during his early childhood. Cohen barked and twitched constantly during his waking hours, and would knock his knee against the inside of the car door. Someone asked his mother once if she had considered an exorcist. His mother was compassionate, but his father did not understand why his son did the things he did, often getting frustrated and punishing him for making noises or other physical tics.

Doctors initially told Cohen's mother that his tics were an emotional reaction to his parents' divorce. Teachers did not understand Cohen; they thought he was mischievous, due to the noises (tics) caused by his TS. His fifth-grade teacher forced him to walk to the front of the classroom to apologize for the noises he made and promise that he would never make them again. He felt humiliated and decided that he would become the teacher he never had, saying: "I always felt like the kid in the corner. I really needed support and acceptance from my teacher and didn't get it. From then on, I knew that I wanted to be that teacher...one who would offer support and acceptance and really be there for each kid." When Cohen was 12 years old, his mother identified his behavior as TS from her own research. She took him to a Tourette syndrome support group meeting, where Cohen realized that other attendees "seemed resigned to a life of defeat" and was "inspired to triumph over the disorder".

In the beginning of his eighth grade year, after Cohen appeared on the Sally Jessy Raphaël show, his middle-school principal decided to let Cohen speak to the school about his Tourette syndrome. Cohen continued to educate people about TS, increasing his confidence and speaking skills.

Cohen graduated from Parkway Central High School in 1992, and was president of the St. Louis Council of Aleph Zadik Aleph.

Cohen attended Bradley University in Peoria, Illinois, majoring in elementary education. During his first week at school he was kicked out of a local fast food restaurant when an employee thought Cohen was drunk and threatened to call the police, even as Cohen and his friends were trying to explain about his vocal tics and TS. Within hours, fellow students began to organize a boycott of the restaurant; the manager phoned Cohen and asked him to come back to the restaurant so he could apologize in person.

After graduating cum laude with many academic honors, Cohen moved to Atlanta, Georgia in the 1990s to seek employment, applying to numerous elementary schools for a teaching position. He interviewed with administrators, but his interviews were always punctuated by his tics. He was rejected 24 times before Mountain View Elementary School hired him to teach the second and third grades. Cohen, who taught about his Tourette syndrome at the beginning of each year, was popular with students; one parent requested his child be removed from his class, but asked to have the child moved back only weeks later. He was awarded the Sallie Mae First Class Teacher of the Year in 1997. Cohen later taught second grade at Tritt Elementary School in suburban Atlanta, before leaving the classroom on the path to becoming a school administrator, supervising other teachers.

Cohen was less lucky in romance, saying, "We called it second-date syndrome: I couldn't get past the first date." Cohen met Nancy Lazarus of Charleston, South Carolina, through an Internet dating service; they married in June 2006 and have one child.

Cohen's book, Front of the Class: How Tourette Syndrome Made Me the Teacher I Never Had, co-written with Lisa Wysocky, was published in 2005. It won the Independent Publisher Book Award for Best Education Book for that year. According to the Pittsburgh Post-Gazette, the book is "well worth reading, giv[ing] insight into what it is like to live with Tourette's...such as when Brad's noises disturbed other people at restaurants, sports bars, golf courses and movie theaters, and when Brad could get a first date with a woman but never a second".

He continued to act as a spokesman for the condition, appearing on The Oprah Winfrey Show on May 26, 2006. He has been featured in a public service announcement for the national Tourette Syndrome Association, and he serves as the vice president for the Tourette Syndrome Association of Georgia. He was a chairman of Relay for Life, a Little League coach, a Major League Baseball mascot, and has received recognition and several awards for volunteerism and community involvement.

The book was made into a Hallmark Hall of Fame TV movie starring newcomer James Wolk...a 2007 University of Michigan graduate...and featuring Treat Williams and Emmy Award-winning actress Patricia Heaton as Cohen's parents; the movie aired on CBS on December 7, 2008. Dominic Scott Kay portrays Cohen as a child.

Following his 2006 appearance on The Oprah Winfrey Show, in 2007 Cohen spoke at a conference after Timothy Shriver of the Special Olympics. According to Shriver, "The audience 'was laughing, then crying, then laughing, then crying, then cheering, and at the end, they gave him a huge standing ovation.'" Shriver suggested the movie to Cohen, and became the executive producer one year later. Cohen was concerned that the movie should stay truthful and avoid sensationalizing Tourette syndrome. He was pleased with the overall result, although he noted some dates in his life were sped up for effect (for example, the date of his wedding).

To portray Cohen's tics accurately, Wolk and Kay viewed videotapes and worked with a dialect coach, describing their preparation as a "Tourette's boot camp". Cohen said the portrayal of his tics is "very, very authentic". Heaton, the mother of four children, was attracted to the script because she related to the strong mother: "Finally, she went to the library herself...as mothers will do to fight for the end for their kids and find out what's wrong...and realized he had Tourettes," Heaton explained.

Although tics steadily decline for most children as they pass through adolescence, Cohen still has frequent and loud vocal tics and barking. He joked: "I'm hoping I don't get kicked out of my own movie."