Do I have a case? What about punitive damages here?

I am not licensed to practice in Georgia, so I will answer based on general principles of law.

Your facts are a bit hard to follow, but I gather that your issue is that apparently you claim your computer was damaged during shipment and UPS refuses to pay the claim for the insurance you purchased. What confuses me is the diagnostic test(s). I imagine shipping damage to be cosmetic (cracks, dents and what not) and not issues with the internal chips. But thats just me.

In the end, if they refuse your claim you will probably have to sue. In this case it sounds like a small claims court affair. In Georgia, small claims court is know as magistrates court and the claim limit is $15K. You could sue for the damage to the PC and potentially add a punitive claim for bad faith (for not honoring the insurance). I am not fully familliar with Georgia law, so you would want to research this a bit. Also, check your UPS paperwork. They may have a forum selection clause in there that makes it difficult. There are ways around that though as meerely defending THAT will cost them more than the value of your claim.

Ultimately, the threat of a lawsuit may make them pay. You can also threaten to report them to the Better Business Bureau, Georgia Department of Insurance and anything else you can think of. They are not likely going to be willing to spend the money to defend the lawsuit. If not, go long and file the suit.