My supercharged 327 ci runs great for a mile or two, but then it becomes unresponsive and backfires. After sitting for a moment, the car will start up and run great for another mile or two. What could cause this?

Sounds like your engine is running too rich and is loading up with fuel. It will seem to run good at first because a cold engine likes more fuel. Once you warm up over a mile or two it will start to run like crap.

All great answers but your true problem lies with your CPU and your fuel system. When you upgraded your car to supercharged, if you didnt change out the fuel injectors or your MAF, this will cause problems with CPU. It is like sticking a square into circle. Most superchargers require 30lb injectors or higher depending on your car. Then you need to have the air flow to support the boost in horsepower. The last thing is to get your CPU to acknowledge, so to speak that you have made these changes and adjust it accordingly (dyno-tuning). If you dont get the CPU to recognize it, it will start and idle and even go a mile or 2, but when the O2 sensor starts sending up signals, it wont recognize the change and therefore wont be able to adjust accordingly, resulting in engine failure.

The problem can be that the diameter ratio of the two pulleys/gears used is not suting your engine, or their is an misalignment between supercharger and engine due to which after running for a while the bearings heat up and creats problem.

Sounds like classic carb problems. Check the float level and all the other carb specs. If your engine is supercharged (not sure what you mean by that) the combonation of performance parts you used may not work well together.

It may be possible that either your timing belt/chain may be slipping or just the fact that your timing may be off s little. Everytime you stop and start it again the computer is resetting itself until it slips again.