There is an idea very similar to this that will work. It has been proven with much higher torques that those seen on a FIRST robot. It is called a CVT (Continuously Variable-Transmission). There are many types of CVT's but the style you are looking at is used on snowmobiles and go-karts.
Here is a thread that shows you what I am talking about.
CVT Thread
Here is a place where you can buy a CVT in the range a FIRST robot would use.
www.hi-lo.com
It would be difficult to build one of these on your own but it definitely can be done. The trick behind these CVTs is that the more torque you apply to the tighter the secondary will squeeze on the belt. This creates more belt tension.
Hope this helps.