I think I may have gotten it fixed for good, but no promises. It was using a static IP before, so we could more easily connect to it if we needed to. I don't think the router had this IP assigned though, so it's possible it was handing out that IP via DHCP to a different device, thus booting the camera off the network. This would explain all the problems we've been seeing with the camera, especially since I have verified that it was still recording video to the SD card during this time, it just wasn't able to upload images over the net.
The camera is now set to just get an IP via DHCP and this should hopefully fix it :)
PS: If you need to connect to the camera to download video files or do other admin tasks, you can do an NMAP scan for the admin port (written on the masking tape attached to the wire of the camera) and you can find the correct current camera IP address that way.