His musical career began with a Bachelor's degree with Honours from University of Winnipeg in 1986. Roberts began performing in a house band for the Blue Note Cafe in Winnipeg under the moniker Bad Brad Roberts and the St. James Rhythm Pigs. The band gradually evolved into The Crash Test Dummies. While studying at university and working as a bartender for The Spectrum, Roberts began writing his own songs and introducing them to the band. After attending a songwriters' workshop with Lyle Lovett at the Winnipeg Folk Festival, Brad wrote "Superman's Song."
Demos of Roberts' songs began falling into the right hands around Canada, and the quirky bar band who had just begun to write original material found itself with record deal options, which led The Crash Test Dummies into a rushed but nevertheless impressive debut album, The Ghosts that Haunt Me. The album introduced the band to the rest of the Canadian provinces, selling over 400,000 copies in Canada alone and also garnishing a 1991 Juno Award for Group of the Year. With more time and finances, Roberts set about to writing the band's second album, God Shuffled His Feet. This to date is the band's best selling and most popular album, taking them into the international arena of musical exposure. The album has sold over six million copies and caused The Crash Test Dummies to be nominated for three Grammy Awards in 1994.
1999 introduced a mix of electronic funk and spontaneous wordplay with Give Yourself A Hand. Roberts met Greg Wells at a songwriters' workshop. Roberts invited him to co-write and assist in recording this album inspired by influences and flavours of Roberts' new home in Harlem, New York.
In 2000, Roberts found himself recuperating from a major car accident in Nova Scotia, where he began jamming with the locals (The Great Wind Jammers from Argyle, Yarmouth County) and produced from these sessions I Don't Care That You Don't Mind. This was the first album released by the band independent of a major record label. As they toured internationally, Roberts described it as "fine as wine", and a "great relief" to record and tour without the pressure of big record label bosses.
Roberts now lives with his wife in SoHo, New York where he continues to write and record his music. He has had many artistic opportunities to work with other artists such as Joe Jackson, Scotty Hard, Greg Wells, Deltron 3030, and Medeski Martin & Wood.
Puss N Boots was released in October 2003 with a European and additional American version. This album brought a more hard-edged sound. Released on October 12, 2004, the acoustic Songs of the Unforgiven was recorded at Sacred Heart Church in Duluth, Minnesota and is described by Roberts as reflective and "hymn-like."
Roberts appeared on VH1's I Love the '90s miniseries. The 1994 episode discussed the song "Mmm Mmm Mmm Mmm" with Roberts claiming that his deep voice is a result of having a third testicle. On the official Crash Test Dummies website, he later noted that this was a joke.
In 2005, Roberts joined the Paul Green School of Rock as a guitar teacher and songwriting teacher.