Born in England to a Dutch mother and British father, Cubitt was six months old when his family moved to Vancouver. While attending university, he worked on a cable music video show called Night Vision, performing in comedic wrap-around segments. Cubitt describes this experience as "the beginning of [his] love affair with acting".
Cubitt continued to study acting full time, until he was kick out of theatre school where he was told he "couldn't act and would never work". Soon afterward, Cubitt landed a leading role in the in feature Alive, starring Ethan Hawke. The film was directed by Frank Marshall, and recounts the story of the Uruguayan rugby team who was stranded in the Andes after a plane crash.
Alive helped spark Cubitt's career and it wasn't long before he landed a series regular role on the hit Canadian series Traders, a show about the high-stakes lives of a group of investment bankers. David starred on the show for three and half years, during which time he won the 1997 Gemini Award for Best Performance by an Actor in a Continuing Leading Dramatic Role and the 1998 Gemini Award for Best Dramatic Series.