The question isn't how many it will run, it's how many displays and of what size are you going to try to run, as well as how WELL it will run.
For example, my old E6600 dual-core with 4GB of memory and 8800GTX ran 5 instances of WoW across two 24" monitors fairly well until I hit Outlands, then performance dropped a little. However, I hit Northrend, and performance dropped quite a bit. While playing lvl 0-60 with 5 instances of WoW, my RAM usage was at 85%. That's with no add-ons, and only firefox running in the background.. with ~3.2GB of available memory in WinXP 32bit.
Once I installed my Q6600, performance had significant gains.. quad core & more than 2GB of RAM is worth it for more than two instances of WoW.

First, follow the advice people have been giving here for over a year:
If you're planning on multiboxing on one computer, stick with a quad core!
If you're planning on multiboxing on one computer, don't skimp on RAM.
Frame rates will be influenced heavily by the size and number of monitors you'll be using.. so you'll need stronger videocards for larger/multiple monitors.

That machine you listed is only a dual-core, only has 2GB of memory, and only has a 9800GT-512MB card. You *may* be able to run 3-4 instances of WoW on one 22" monitor with decent fps, but those windows will be tiny. The first limit is the memory. Get 4GB - it's cheap enough now. ($55 for 4GB of Corsair DDR2-800 at newegg.com.. and nearly anyone can pop in new memory.) Second, I'd take the Q6600 quad core over an E8000 series dual-core any day for 4-5 boxing. A Q6600 is $190.