More to do with total car...
Basically, whatever the "total" car can handle. What I mean by this is Tim Allen's Mustang (Casper) referred to previously had LOTS of power but the chassis and suspension (let alone tires) were not up to handling that power.
But, a car can be built to handle 600-700 hp (not sure how close to a steet car that would be, though). Another classic example of what I'm talking about is the M3 vs. M5. The M5 has 400 hp (almost 70 more than the M3) but is built to handle it (wide tires, suspension calibration, and is one heavy mother!!!!). 400 hp in a Miata would be dangerous.
I modified a '92 Mustang LX 5.0L to the point of 350 hp - for this car and the set up I had, this was and ideal street car. Any more power and I would not have been able to keep it on the ground. My uncle had a Mustang he modified in almost the exact same way and then put a blower on it. He was pushing over 550 hp with it and said it was the biggest mistake he had done with the car - he would break the tires loose in 3rd gear at highway speeds simply by punching it - not safe (and not enjoyable, either). But, if you take that 550 hp and put it in a AWD Porsche or a tuned Viper, that might be more acceptable.
I guess it comes down to power to weight ratio, chassis stiffness and suspension set up....
I agree with "ff" on the point of it not being the driver. A 50 hp Yugo can be dangerous in the wrong hands...