I've experienced this peculiar set of problems from time-to-time at a few sites over the years, particularly the intermittent "network cable unplugged" pop-up messages. I discovered that the solution was to perform a deep power cycle of the effected computer(s).
Additionally, using a good power bar helped to prevent further repeats of the same set of problems for most of the computers (I show a picture of a really good power bar in my answer to the question " Is it bad to leave your computer on during a thunderstorm? ").
To perform a deep power cycle:
- Shut down the computer as usual
- Physically un-plug the power cord
- Use the power switch to attempt to power the computer on (this will fail, but you might see the LEDs blink on the front for a moment as any remaining power is extinguished)
- Physically plug the power cord back in
- Power on the system
If this resolves the problem for you, then you may have some power fluctuation issues at this site. A power bar like the one I suggested in my answer to that other question seems to help at the sites I've noticed these problems at (I used an AC volt meter to measure the power and discovered different power levels at different times of the day; apparently the local hydro company says this isn't a problem because the variations are within acceptable limits), but if they vary too much then a battery backed UPS unit may be a good solution (even if it only provide a few minutes of standby power, what you're after is the consistency of the power levels).
Of course, making sure your system and its network drivers are up-to-date is always recommended.