So, I bought some cheap switches and..
Not-so-long story short: we had some very old 10/100 5 port switches sprinkled around the place and I thought I'd replace them with some cheap (Netgear) gigabit ones. I know, I know. Anyway, the first sign that things weren't fine and dandy was when the network slowed to a crawl. Suddenly network printers were showing as offline, with some printers freezing mid-job. DHCP was failing on wireless access points, and so they were rendered useless. After manually setting the speed on the core switch to full duplex gigabit for the connection, everything fixed itself.
Now, I'm not going to keep these switches installed, but I'm very curious about what someone with more network savvy than myself thinks may have caused this, because whatever these things were doing, they were screwing with equipment not connected directly through them - anything connected to the same core switch as that Netgear junk would be rendered useless. I've never seen anything quite so catastrophic, and honestly can't imagine what they were doing to cause such havoc.
Any ideas? The switch(es) are Netgear GS205, for anyone that wants to avoid them like the plague.