It turns out that the problem was the network card-- the cablemodem didn't like it. Switching that network card with the one used for the internal network (Tulip chipset) fixed everything.
Thanks to everyone who helped-- even the responses that weren't directly responsible for the solution gave me a better understanding of the problem. My next issue now, which isn't that important... To find a way to limit the amount of data apache can transmit at any time (I transfer large files back and forth through it, and whenever I try to get something too large, the network is effectively killed for everyone else) Thanks again! Mike