Try setting up internet sharing and seeing if all your macs are able to share with the mac pro in question.. If they are, then your ethernet ports are working fine.. aside from that, try a PRAM reset as that will reset the ports..
The last resort would be to use Apple Hardware Test to test your ports or Apple Service Diagnostic which I have if you need to really troubleshoot your ports.
Try to remedy the situation with internet sharing tests and PRAM reset.
There are a few other tools available on your Mac that can help troubleshoot.
Network Utility pulls a collection of command-line tools into a pleasant GUI for ease of use. The Info pane, when set to the active Ethernet port (typically en0), shows the speed at which you actually connected, the number of packets sent and received, and a number of error counters.
These error counters should be zero. Worst case, a handful of errors that happened at startup, and almost no more, ever. Anything else indicates a bad interface at the Mac or (more likely) at your Switch, or (even more likely) a bad or marginal Ethernet cable.
I have exactly same problem My Mac Pro 3.0 looses ethernet connection once in a while. I have checked all i can figure out. Modem/router, cables, with and without NAT and DHCP and that problem just won't go away. I made NVreset, i even changed my Bewan modem to Netgear DGN3500 (to get N-wireless and gigabit ethernet). I've had this problem for a few months. I believe it started after some updates. I have Macbookpro and one Shuttle (ubuntu) pc on that same router but they dont have any kind of problems with connectivity.
To get it working i have to pull out Ethernet cable and then i just plug it in and everything works for a while. Sometimes even days. I doesn't happen if i'm working all the time. Usually it happens when my computer has been on but nobody hasn't used it for one hour or something like that. I checked that there isn't loss in packets (network tools). Ethernet diagnostics is almost allways showing that everything is ok. Still my browser doesn't load pages. If i keep trying for a while then it might suddentlywork again. Somehow it feels that those ethernet ports are in sleep. Both ports does that. Very irritating...
Storms? Is the modem and router both behind and filter/surge protected and have uninterrupted power?
Heard of a G5 that did lose an ethernet port.
And worst case, try picking up a PCIe ethernet card from OWC, I know Sonnet use to have one and should still.
When I talked to APC they had two things to say, yes I should filter/surge coax first before it gets to the modem, then go back through and use RJ4 port on UPS again before going to router, as the first can still allow more than you want going downstream to a computer.
coax => RS1500 coax surge port (in/out) => modem => back to RS1500 RJ45 port and out => router
2nd item: that APC RS1500 can cause trouble (as I found out and as did person I talked to), but have never had trouble with UPS from Cyberpower - and have a number of them in house for each PC, computer, TV and other equipment now.
That though is to prevent accidents and also storm from disabling in whole or part of damage that can occur to modems and routers, and to ethernet ports.