It's possible but it kind of sucks. On most computer keyboards there are limitations as to which keys can be held down simultaneously (and how many), and the keys are on/off - there's no pressure detection. Also, computer keyboards are pretty small - you can't fit many notes on at all - and the keys are not arranged regularly.
It is possible with VMPK, and some people have shown they can play great music on it (https://www.youtube.com/watch?v=LhfSF_RpxjI) but I don't know how hard it is to get to a stage where we can be productive. It would be helpful if you have an NKRO keyboard.
I've always wondered if its possible to use a standard computer keyboard for this sort of thing. What are the main reasons that this wouldn't be possible or likely?