Dunham was born in Dallas, Texas on April 18, 1962. When he was three months old he was adopted by real estate appraiser Howard Dunham, and his homemaker wife Joyce, who raised him in a devoutly Presbyterian household in a rich Dallas neighborhood as an only child. He began ventriloquism in 1970 at age eight, when his parents gave him a Mortimer Snerd dummy for Christmas, and an accompanying how-to album. The next day he checked out a how-to book on ventriloquism from the library, and explained in 2011 that he still had it, remarking that he was "a thief in the third grade". By the fourth grade, Dunham decided he not only wanted to be a professional ventriloquist, but the best one ever. Dunham began practicing for hours in front of a mirror, studying the routines of Edgar Bergen, and the how-to record "Jimmy Nelson's Instant Ventriloquism", finding ventriloquism to be a learned skill, similar to juggling, that anyone with a normal speaking voice can acquire. Dunham explains that as an only child, he enjoyed being alone, likening his solitude to a "warm blanket" with which he could explore his own thoughts and ideas, which prepared him for the solitude of living alone when he later moved to Los Angeles as a struggling comedian.
When Dunham was in the sixth grade, he began attending the Vent Haven ConVENTion in Fort Mitchell, Kentucky, an annual international meeting of ventriloquists that includes competitions, where he met Jimmy Nelson in person. Dunham has missed only one ConVENTion since then, in 1977. The organizers of the ConVENTion eventually declared Dunham a "retired champion", ineligible from entering any more competitions, as other attendees were too intimidated to compete against him. The Vent Haven Museum devotes a section to Dunham, alongside Señor Wences and his idol, Edgar Bergen.
Dunham began performing for audiences as a teenager, in various venues such as school, church, and during his job at Six Flags. By his middle school years, he began to perform for banquets attended by local celebrities such as Dallas Cowboys quarterback Roger Staubach, having developed his style of lampooning those he performed for, using the puppets to say things too risque for him to say without them. Dunham's television debut came in 1976 when the still prepubescent performer caught the attention of Dallas reporters like Bill O'Reilly, who interviewed Dunham for a local news story. Dunham later did commercials for Datsun dealerships in Dallas and Tyler while in high school. While emceeing a high school talent show, he dealt with a heckler, and won over the rest of the audience. During this period he became so associated with his craft that he and one of his dummies "cowrote" a column in the school paper, and he would pose with his dummies for yearbooks as an inexpensive way to acquire professional photos of his act for promotional purposes. He was voted Most Likely to Succeed, and in 1980, after he graduated from high school, Dunham gave himself a career goal of obtaining, within ten years, an appearance on "The Tonight Show with Johnny Carson", which was seen as the "holy grail" for comedians.
That year Dunham began attending Baylor University, hoping to graduate with a degree in communications, while performing around campus. He would also fly around the country on weekends, doing up to 100 private shows a year, entertaining corporate customers such as General Electric, whose CEO, Jack Welch, he mocked during his routine. By his junior year in college (1983–1984), Dunham was making $70,000 a year, and as word spread of his act, he landed featured spots opening for Bob Hope and George Burns, though he still perceived his act as raw, as he did not have any knowledge of standup comedy beyond his Bill Cosby albums. He caught a break in 1985 when he was asked to join the Broadway show "Sugar Babies" with Mickey Rooney and Ann Miller, replacing the outgoing variety act. For the naive and devoutly-raised Dunham, Broadway was a new world filled with beautiful showgirls and crusty stagehands, and his first taste of entertainment industry egos came when Rooney called Dunham into his dressing room, and told him he was there for one reason alone: so that Rooney could change his costumes. Dunham also performed at the Westbury Music Fair on Long Island. These early experiences, in which he used characters like José Jalapeño on a Stick, taught him the value of modifying his act regionally, as the jalapeño jokes that worked well in Texas were not as well received by audiences in Long Island.
After graduating from Baylor University in 1986, he continued honing his act in comedy clubs in the Southwest with new characters such as Peanut and Jose Jalapeño, but struggled against the perception he relates from fellow comedians that he was not a true a comedian because he relied on props. His experience at Catch a Rising Star in New York City served as a bitter confirmation of where ventriloquists stood in the comedic food chain, as the emcee at that club gave Dunham little respect. According to Dunham, after he arrived at the club in the evening and informed the emcee that he was a ventriloquist, the emcee reacted with derision, telling Dunham that he would be given a late time slot, and after that time slot came and passed, kept postponing Dunham's stage time until Dunham left the club. By the end of 1988, Dunham felt his career went as far as it could go in Texas, and he moved to Los Angeles, California, never having, as he has commented, "a real job", much to the concern of his parents, who assumed he would relegate his act to local venues such as church groups. When he first arrived in Los Angeles, the comedy in his act bombed. Dunham attributes to his underdeveloped comedy, explaining that while the characters' personalities were developed at that point, his jokes were not. In addition to this, the comedy world was not welcoming to ventriloquists, and his manager, Judi Brown-Marmel, did not use the word "ventriloquist" when finding bookings for him, choosing to present him as a comedy duo. After Dunham became friends with Mike Lacey, the owner of The Comedy & Magic Club in Hermosa Beach, Lacey gave Dunham a steady slot at the club, where Dunham sharpened his act by observing the techniques of comedians like Jerry Seinfeld, and taking the advice of colleague Bill Engvall, moving away from his G-Rated material toward edgier, more adult themes.