This comes from personal experience...
In 2008 we got 'too greedy' with our wheel design and made the spokes way too thin. I think each spoke was something like .167" thick.
While practicing between LA and Atlanta, our driver turned a corner and something shinny rolled off our robot. After finishing the lap I went to go an examine what came off our robot, only to find that the wheel hub was still on the robot, but the rest of the wheel was gone.
After looking at the rest of the wheels, I found that others were starting to fatigue as can be seen in this
photo.
The problem was caused by side loading and how thin our cross sectional area was. When we were doing our pre-fabrication analysis using Cosmos, we only checked the compression strength and 'normal' direction loading on the spokes. In both cases, the wheel had a factor of safety of >2. After we began breaking wheels I did another analysis on the effects of side loading and found that our FoS was <1.
If you look at the picture I linked, all the wheels broke at almost exactly the same spot, which also happened to be the smallest cross section of the spoke.
For Atlanta, we made a new set of wheels that had a thicker spoke and took the weight hit since we had 8+ lbs to work with.
Yes, weight is an important factor, but you don't want to under-build your wheels and have them break in the middle of competition. There are other places you can save weight with out sacrificing the structural integrity of your wheels.
Alan is right, where the radii are at now only help in the 'normal' direction. When it comes to side loading, they don't do anything.