Hello Garth...
NIC
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NIC = Network Interface Card -- Your ethernet card or port. If you have multiple network interfaces, the idea was to try disabling all except the one you are trying to use. This will ensure there are no packet routing problems. In your case, your dial-up connection is one of the "NICs". Other "NICs" could be a wired or wireless ethernet interface. Disable them "Network Connections". But, shouldn't be the problem because routing and DNS continue to work.
PING AND NO BROWSE
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Yes. There is no DNS resolution problem because you can ping the domain name when the browser reports "server not found". Something else is the cause.
IE BROWSER CACHE?
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Running out of ideas that don't contradict other things...
Temporarily ignore that FireFox also fails.
Let's review: Network routing still works. DNS still works. Downloads continue after browsing stops stops working. This points to the browser itself as the problem.
It is as if port 80 (web browsing) is allowed for 3-4 minutes and then disallowed.
When IE cannot connect to a server, it sometimes does not report it right away and will display a cached copy of a page. To prove IE is getting fresh pages during the first 3-4 minutes, go to a website with frequently changing content like CNN.COM.
WEBSITE PROBLEM
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Running out of ideas: Is the website the problem? Proof: Try to browse multiple websites.
DENIAL OF SERVICE ATTACK?
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Out-of-the-box: If your PC were being attacked with a large number of packets as in a denial of service attack, web browsing might fail in this way but small quick things like PING might not. The attack might not start for 3-4 minutes hence the correlation with time. After the browsing fails, check your PING response time vs. before the failure (would be higher) and your connection "packets received" count on the "Status" window (would be rising continuously).
-----Steve