A similar thing happened with my parent's iMac. They had a G5 iMac, that had many documented problems from the day they bought it. Every time it went in for repairs, it came back in worse shape. About a year and a half after they purchased it, and after one repair too many (that broke something else) my dad get fed up, and got in contact with a VP of customer service. The same day, the local Apple store called them told them to come pick up their brand new 20" C2D iMac, and drop off the old broken machine. They transferred all the files over, etc.. That machine has been working great ever since. (it probably didn't hurt that my family has purchased an Apple computer of almost every-other generation since the Apple IIgs, from the IIgs, to an SE, Centris 650, Quadra 840av, numerous laptops, 3 iMacs, etc..).
My mom decided to upgrade the RAM, and add on Applecare.
So I think it really depends on the instance. If 3 years post purchase your screen dies out of nowhere, Apple probably isn't going to replace it. If, however, you have had it in for repair numerous times, and it has never worked correctly, they are much more willing to "make things right"