Can't ping but can access internet just fine?

A lot of sites may choose to turn off ping response for security reasons, but are still accessible through the limited number of ports that are opened (like 80 for http and 443 for https). Here's some documentation for Linux: http://www.linuxhowtos.org/Security/disable_ping.htm

So in summary, being unable to ping something doesn't mean that you have network issues.

Hope this helps!

It turned out to be the cable modem at fault, and a hard reset of that component seems to have straightened it out. In retrospect, since I removed every component of my network except the cable modem and a single laptop to simplify the test, it probably should have been more obvious at the time that it was the culprit.

Your router can filter ICMP-packets from external world. If you can login to router and try to ping from it's shell it can give you some hints