A DSLR is a tool that can get the job done.
It's really simple: if something can shoot video, it can shoot a movie.
But not all equipment have the same imagequality or control over the image.
So your cellphone is less usable to pull that job.
To me the big plus of a DSLR is the ability to shoot in lower light conditions than a lot of 'camcorders'. When my EX1R is blind, a 5D can still produce an usable image.
The downside is that rolling shutter in DSLRs is really bad and that in certain situations the image is either soft or suffers from moire.
(I must tell you that I make videos for a living; I prefer a C300 over any DSLR for shallow DoF or lowlight shots. But in my case the client pays for the camera. I just can't say to a client: "sorry about the jello effect; that normal with DSLRs"
)
Having said that: yes, you can make a (good) short with a DSLR.
Beware of the big pitfall: unfocussed shot because you use very shallow DoF.
BTW, your first ultrashort should be a video of a camera taking a picture when pressed by a carrot. Just to piss that one person off