I've gotta say it sounds like you're talking about a DNS problem more than it would be a firewall/router problem. You might want to compare the two computers DNS settings and see if they're different, try manually pinging the addresses and see if they correlate and if you're running Win2K or WinXP try the command ipconfig /flushdns from a command window to clear the DNS cache on your computer.
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Halfway to hell and picking up speed.
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