Have you verified the amount of Syslog traffic you're receiving? Use Wireshark or another packet sniffer to see how many SysLog messages you're receiving. A single device forwarding Syslog messages could be overwhelming the SysLog server - it has less to do with the number of devices and more to do with the amount of Syslog traffic each device is sending.
If this is the case, you may be able to use Windows Firewall to block UDP port 514. If blocking the port solves the issue, you may be able to set up some filtering rules to help manage the load and then unblock the port. However, if there's a high volume of SysLog messages, I would recommend either Kiwi Syslog or the Orion Syslog server, both of which are better-suited to handling higher volumes of SysLog messages.
If you're not being overwhelmed by Syslog messages and the Syslog server is still behaving badly, let's try to set up a more in-depth troubleshooting session. Let me know what you find out and we'll go from there.
It looks like this was the case - since by default the syslog server records every type of message sent, it was being buried in events from a couple of devices. I took the box off the network in order to stop the traffic then configured the alerts I wanted to see and things seem to have settled down considerably.
Thanks very much for the help!