Moodle may show them as coming from the same IP (All my external users show as coming from the same IP which is an upstream firewall) but that it just NAT doing its job. If the machines have the same IP on the same network it will not work.
In a local LAN situation all computers should show with their own and individual IP.
These are not Moodle restrictions. They are IP network protocol restrictions. If everything else (network wise) is working then I suspect that the computers do not have the same IP and that the problem is somewhere else.
I don't know about on a Linux workstation, but on a Windows machine I can not have two different logins to Moodle at the same time. As soon as I open a new browser and point to Moodle.org it identifies the cookie from the first session and shows me as logged in. Logging out in either one also causes all sessions on that computer to log out (may need a refresh to show).
I think Ben has it right. AV software or an overstrict settings on the browser is the most likely cause. Some AV programs will stop the loading of cookies which will stop access to Moodle. This is the first thing I would look at.
Hellen, either way, I hope our comments help will finding a solution to your problems.