What do you mean by “booted”? As in, you lose your connection to the router?
Assuming it works perfectly fine if the computers are not on, i.e. there is no physical problem with the cable or the connectors on either end, then possibly:
a) you have configured the same static IP address on your PS3 as you did on one of the computers
b) you have limited the number of IP addresses that your router will assign from the DHCP pool
Option A: you should know the answer to this right away, either you configured static IPs or you didn’t. If you did, check the IP address you configured on each machine and the PS3, write each one down, and make sure none are the same. If you didn’t, and all your devices are configured for dynamic/automatic addressing, then this is not the problem.
Option B: I can’t imagine any router would come configured out of the box to only allow maximum 3 DHCP leases, but I have seen stranger things. You’ll have to access the configuration of the router (via the router’s IP address, from a web browser on a computer) and somewhere in the Local Network settings you should find the DHCP options. Make sure the starting and ending address in the pool aren’t really close together (e.g. 192.168.1.100 - 192.168.1.102)
Please check those two and report back.