Swap the machine/proxy assignments to see if it's a problem with the machine, or the proxy.
If it's the machines, then you'll have some difficulty tracking it down, but since you can see the HTTP requests using fiddler you should be able to see what the difference is in the requests that causes the site to spit you out.
If it's the proxy, check to see if both proxies are configured identically. If not, swap the IPs on the proxies to find out if it's a problem with the IP or the proxy configuration. Pay attention to the HTTP request to see if perhaps one proxy is giving you the 403, rather than the site itself - could be blocked at the proxy.
If it's the IP, and nothing else is different, then email team@stackoverflow and request their assistance in determining if they have that IP blacklisted.
If at any time in the past your computers abused SO (lots of traffic, bot, etc) then you may have fallen victim to such an IP block that usually automatically clears. Are there others using the proxy? Could be them. Either way, if you're certain it's the IP then email the team and ask if they can give you an idea why this might be happening. They won't spend much time on the issue, so don't expect a thorough examination, especially if you haven't done most of the investigation already, but they are usually good about answering simple requests.