I imagine any animal can die of fright, some are just more susceptible than others. Extreme fear and stress can cause stress myopathy which will cause the heart to stop. I've had it happen at work. It's not a pleasant thing to be examining a bird and have it die in my hands. Sure, it may have had a broken wing but the exam is sometimes too much for it no matter how quick or gentle I am.
What kind of scare did the rat have? Or is this just a question out of curiosity?