Since all the computers do work, albeit not always at the same time, I would be looking for environmental issues at the hardware level initially, as this is often overlooked.
First of all, what speed are the LAN connections? Are they coming up at the same speed when this problem occurs? (Does the Cisco have connection lights you can watch?)
Are the network cables within the length limits for twisted pair at the LAN speed you are using?
Are all the computers running off the same phase of the mains electricity supply? I have seen issues caused by earth leakage currents through the mains supply due to machines being on different phases causing high currents to flow through the network cabling. This really messes with the signal to noise ratio in the cabling.
A quickish way of checking is to disconnect half the machines and see if the problem goes away. Then try with the other half. Then try with three quarters of the machines rotating by quarters to see if a specific machine or block of machines are causing the issue.
If not hardware, it may be worth running something like Wiseshark to determine the difference in communications between a successful connection and a failed connection.