Damn. I'm jealous. Wish I had all that hardware to play with 
60 to 62 C for your existing dual 5850 setup is a very good temperature range. I fear that if you put 4 of them next to each other like that you will had some major problems with cooling them. First off heat will bleed into the card next door (typically the one above it) and the resultant effect will be that the top card will be baked like a cookie. Second I'd recommend you leave a nice gap between the cards. The fans need some space to intake air.
I can't quite tell from that picture but are those 5850 using a turbine cooler or a non reference design (ie aftermarket)? If fans instead of turbine then I think you're asking for trouble putting them so close to each other with no room to breathe. If turbine you may be able to get away with it if you maybe I dunno jam something in between them to open up at least a tiny gap. Also if you do this put two high CFM rated fans at the back where the intake vents are and run them at full blast to ensure that cool air is force fed into them.
If you can get each card to run without going above 80 - 90 C you're golden
60 to 62 C for your existing dual 5850 setup is a very good temperature range. I fear that if you put 4 of them next to each other like that you will had some major problems with cooling them. First off heat will bleed into the card next door (typically the one above it) and the resultant effect will be that the top card will be baked like a cookie. Second I'd recommend you leave a nice gap between the cards. The fans need some space to intake air.
I can't quite tell from that picture but are those 5850 using a turbine cooler or a non reference design (ie aftermarket)? If fans instead of turbine then I think you're asking for trouble putting them so close to each other with no room to breathe. If turbine you may be able to get away with it if you maybe I dunno jam something in between them to open up at least a tiny gap. Also if you do this put two high CFM rated fans at the back where the intake vents are and run them at full blast to ensure that cool air is force fed into them.
If you can get each card to run without going above 80 - 90 C you're golden
I am not sure if they're reference or not, they're Sapphire Radeon HD 5850, specifically, Sapphire Radeon 5850 on Tiger Direct. They appear to have a fan, not a turbine of any type. I can't tell if they take air in the back or exhaust out the back. I'm looking at a rig now, thats been running for a while and its at 66C on both GPUs and one of them (fans) is at 75% other at 53%. The one at 75% is because it is just as close to the PSU (on bottom) as these 4 cards will be to one another, but then again.. they're not that close together to each other and I know the GPU puts off more heat than anything else in the case.
I might test it.. but I might just put 3 in still, or perhaps only 2 and just be stuck with a spare card in another machine alone, until more come in.