It might be a local DNS issue depending on how your router handles loopback traffic.
You might need to add your domain name to your hosts file and point it to that computer’s IP address.
Wordpress acts funky when the name is set to “x” and you try to access it at “y”. Usually though, the site will come up, but look strange because it can’t find the CSS to draw the page correctly. Also, links won’t work because they’re hardcoded to that address.
This doesn’t seem like a new site though, so I imagine that it HAS been working from the computers in question before. Has anything changed in your network setup recently?