Interesting problem.    Let me give you a bit of insight into my troubleshooting process.  Keep in mind that I'm primarily a network guy - If I can ping it and I know the ports are open, it's a server or a client thing, which I don't usually get into much anymore.  
I always do this kind of troubleshooting by the layers of the OSI model.    Easier that way, and you know you've covered all the bases.
There's a few questions, however.  I'm rusty on my Macspeak, but..  Are you using straight IP, or is there any Appletalk involved?      When this computer "breaks", how do you fix it?  
1:  Physical layer - Cabling 
You've tried another patch cable and switch, so you know that's not the problem.  How about plugging the machine into a totally separate jack leading through the wall?  Tried another machine on that same wall jack to see if it does the same thing, to tie the blame to the cable?  
Do you have a link light after it's dead?  Do you see the same kind of traffic activity blinking (assuming that you do have a traffic LED on the NIC) after it's died than when it's working?  
2:  Data link layer (NIC<->Switch communications)
Look at the speed/duplex settings on the Mac and the switch.  Make sure it's all at auto/auto.  If it is, try locking everything down to 10BaseT, half duplex.  Yeah, it's slower, but he'll probably never notice.  You might be getting a NIC that's having errors due to speed/duplex problems and either the switch or the NIC is shutting him down to prevent it from corrupting the rest of the network.  This is more likely if it happens when printing, as that usually sends a LOT of data across the network, typically WAY bigger than the ordinary size of the file.  Lots of traffic could mean lots of errors which means it gets shut down.  
A big FTP or file transfer might also help identify this.  Try to find a file that's 200+MB and transfer it from the server.  
4:  Make sure that you have one "main" switch that has connections to all the rest of the switches and to the file server.  Make sure you don't have switches cascade one to another to another, except where absolutely necessary.  Never go more than 3 cascaded switches deep.     More than 5 switch hops is bad.  
If you can scrounge up a plain 10BaseT hub, try and put the hub between the Mac and the switch.  That should elliminate speed/duplex problems.  Not a 10/100 or any kind of "smart" hub or switch.  Just a dumb little hub.  
Make sure that all "fancy" features are turned off on the switch port it's plugged into.  For example, a Cisco port will go through it's speed/duplex check, then try to look for an EtherChannel, then make sure there's not a spanning tree loop, THEN go into active mode.  This often breaks DHCP on Win9X machines, as they give up trying to get an address before the switch port goes active.  
Any chance you've got a bad NIC in this box?  Can you change it, or is it onboard?  How about snagging a firewire or USB NIC to try and see if it makes a difference.  
Layer 3/4 - Network (IP communications)
1:  Change it to a different static IP and see if that helps, to rule out someone ELSE with the same static IP plugging in occasionally.  
2:  Setup a ping -t from a Windows machine to do a constant ping to the device.  next time he has a problem, check and see when it stopped.  Better yet, use a free ping checker app like Ping Plotter to watch the success rate.  If this continues and his server access stops, then you've got some kind of client/server problem.  (Not Mac bashing, just that the network is OK).  If you CAN still ping it and he's still broken, look at your ARP tables to see if you've got a different MAC address.  If so, you've got another machine using the same IP.  
Try and setup a similar ping from the mac to the printer he's sending to.   If the ping stop when his system looses network connectivity, it's probably a Layer 1 or Layer 2 problem. 
As TG suggested, see if you can ping anything or browse the Internet after the communications die.  
Best of luck!  
- G