The results of our morning network downtime are in and the problem was not with the NetApp. Thank you for all of the suggestions. Since I didn't have any replies from someone saying, "Yeah we had that exact problem and we did this to fix it," I had confidence that the problem was with our Cisco configuration and it was. Actually there was one person that had a similar result when their default route was entered incorrectly, but that was not the case for us.
We are using a bridge-group on our Cisco router to forward broadcasts and allow Sun Autoclients to boot across the router. There was an access list to limit these forwarded broadcasts to packets with the Sun manufacturer ID in the hardware address. This access list was also preventing the NetApp from reaching the router on its own subnet which was also part of the bridge-group, probably because the router was dumping its broadcasts since they didn't have a Sun manufacturer ID. The fix was simple, we added the NetApp and other manufacturer IDs to the access list.
Now go back to your partying at LISA...
-- Mike