Well doesn't sound like an idea situation, if the pci-e slot is so close to the bottom of the case are you sure a 2 slot card will fit in there, also is the case long enough for these cards, the 470 and 5850 are like 7, 8 inches long and the 5870 is an inch on that. Cooling will not be great at all, even if it does fit in the card would be trying to suck air between card and the case so airflow will be at an obsolete minimum.
But you should be asking yourself, can my computer handle a card like the 470 5850, and the 5870. What are the rest of your computer specs and what resolution are you playing at
The rest of the computer's specs are pretty good. Core i7 950, 6GB ram. As for resolution, currently it runs kind of low due to the card, I do run widescreen, and don't necessarily need to go to true 1080p but anything higher than what I'm running would be nice. I usually run 1280x720 or 1440 x 900, depends on the game of course since the demands of the game kind of dictate whether I'll even be able to run that high.
The card that's in there now is a Radeon HD 4850, and it's pretty long, I think it's even the same length as the 5850.
Problem with transferring everything to a new case is that computer is under warranty, if something breaks I'd much rather swap out the power supply / video card to the originals and send it back then swap out the entire case to send it back, then get the fixed computer back and swap it all back again.
Also, sorry for the noob question, but when you say dual slot, do you mean I need 2 PCI-E slots? I've seen some pictures of the physical hardware and it certainly doesn't look like there's 2 PCI-E connectors.
Or just that it's fat and will cover a second slot, without actually using it? If it's the latter, then it should fit fine, as there's nothing above it.
It just means that it's fat pretty much and will over 2 slots, and it'll need two expansion plates on the back to fit. If you have a 4850 then most likely it's a dual slot, and you'd probably have enough room for those cards if there still some space long wise in your case.
So overall you could put one of those new cards more than likely, but i wouldn't think about overclocking it, as it'll probably run fairly hot with the restricted space