The network has on the order of 80 computers attached to it, none of which have any speed or disconnect issues that we are aware of.
Tech support got back to me and asked me to directly connect the NAS box to a computer to see if it would hold a connection. This seemed like a reasonable thing to do to me. In preparation for this, I changed the NAS back from DHCP to a static IP address. On a whim, I tried my large file transfers to it again. Things have been running stable now for more than 24 hours whereas before I was getting disconnects every 10 to 15 minutes. I made no other changes to the NAS box. Its physical connection to the network is exactly the same as it was before. In fact, I never even touched the box. I simply changed it from DHCP to static IP using the web interface.
I do not know why switching from DHCP to a static IP would make a difference in network stability. The DHCP lease was good for a week, so it should not have been trying to renew every 15 minutes. It appears that something, though, when running in DHCP mode is messing with the NAS network connection every 10 to 15 minutes. If anyone has a technical explanation as to why this might have been happening and why switching to a static IP address solves the problem I would love to hear it.
To the original creator of this thread -- are you using DHCP? If so, try switching to a static IP and see if all of your issues go away. Good luck.