Ever since the release of the Canon 7D I've been trying to understand the differences between the 7D and the 5DM2. In the process, I ran across Stu Maschwitz's (visual effects supervisor for Sin City and The spirit) blog ProLost.
Stu has a great post that goes into detail on the 7D and how it compares to the 5DM2 and more importantly how the 7D's APS-C sensor is close to the Super-35 sensor size regarded by filmmakers as a holy grail for digital cinema.
Even though Stu's post is talking about the 7D his insight on sensor size is not vendor specific. It sounds like APS-C sensors as in the 7D, D300s and other small sensor DSLRs may be a near optimum size for digital video.
So the bottom line is, no, you don't need a full frame body for great video. But the advantages are as Stu says, "...the twice-as-vast area of the 5D Mark II’s sensor means more flexibility with more lenses, and more control over DOF. Yes, this includes the option to push it to a fetishistic extreme, but also the ability to achieve cinematic DOF with a slower stop of an affordable, flexible zoom lens".