This was all supposed to be fixed in Sony's upcoming 1.6 firmware update. But, is it already broken before its release? Apparently so. The Inner Bits has "obtained" info about the upcoming firmware revision, and it's not looking good. The background downloads are limited to while you are in the XMB. If you launch a game, the system will pause your download because "the PS3 needs extra system resources (primarily memory) to do background downloading."
If this is indeed true, it certainly can't be due to the system's inability to play a game and download a file. Although Sony's system has the hardware to deliver the goods, it seems Sony is just not creating user-friendly software to take advantage of the system's "potential." Will we have to wait for yet another firmware revision for Sony to get this right?