You probably got a remanufactured (re-man) one, not a new one. At least for iPhones, if they can't repair, Apple care gives re-mans. These re-mans are identified by their part number. As a result, Best Buy wouldn't take my iP5 as a trade in because they could tell it was a re-man. Which I think stinks on Apple's part. I was paying for extra insurance, I should get replaced like for like, not be saddled with the scarlet letter for a re-man.
tl;dr - Apple Care does not give out new units for claims. Consider that in your calculus for your personal value assessment of Apple Care.
This stinks on BEST BUYS part... not Apple. I've had several reman'd iPhones and they've all been just fine. I've had several brand new ones that have had issues right out of the box. I don't think you're any more or less likely to get a "bad apple" (pardon the pun) with a reman versus a brand new in box. Apple refurb process is presumably better than anyone else's because they have the knowledge and tools necessary to do it right.
I think it's shady on the part of BB to not take a reman'd phone. Who cares if it's been reman'd? So you could go to apple and have the screen, home button, battery etc. replaced and BB wouldn't give a hoot but if you get a whole new phone with those replacement parts, BB cares? Time to stop doing business at BB...
Really? I thought they were new! The heck!
It's also my understanding that they give you a new body, new screen, new battery and the only things they reuse (after testing of course) are the high dollar items like the processor and other internal components.