How can I diagnose why I can no longer access certain sites on certain networks since I upgraded to Mountain Lion?
When I've had this issue in the past, like the two day period when I couldn't access Flickr from my laptop but I could from other machines, the solution was to flush the DNS Cache: essentially a small database on your machine that can help (or so my understanding) speed up network access.

According to the latest Apple Support document on this, the command is:

sudo killall -HUP mDNSResponder


http://support.apple.com/kb/HT5343