You might have ARP issues. That's the table routers store in memory to connect hardware (MAC) addresses to IP addresses. On most switches and routers there's a command to view and also flush the ARP table and let it rebuild. If you can't access the switch directly via some kind of control interface, you may have to reset (hard boot). I believe your common linksys has a recessed reset button, but could possibly destroy your config, so make sure you write everything down before trying that. I can't recall ever having a problem rebooting Cisco routers.
Now, if it is a DHCP problem, it sounds like the DHCP server is confused with the hub and sees it as one machine. I've never seen this happen, but seems probable. So could be a server problem, but it seems more likely that it's the hub either not relaying hardware information correctly, or the switch not updating it's ARP table fast enough.
Answer: check "ifconfig -a" on unices (I think Apple also) or "ipconfig /all" on windows to get a glimpse of mac and ip settings, then DHCP settings on the server (which may be your switch) for lease list and lease times and maximum connections, stuff like that; make sure it's doling out the correct subnet mask -- too large of a mask, the fewer addresses you acquire. Check the ARP table on any switches between you and the DHCP server. If you're connected to an ISP, then you might be only able to use one or a handful of IP addresses, unless you are using NAT.
One last thing. I highly doubt it, but it could be a clash of networks (i.e. you have neighbors somehow sharing your network and vice versa).
Hard to diagnose on this site and sorry for the overloaded answer, but good luck. I sort of went backwards; from complicated to simple, lol.