The first thing I'd try to do is check to see if the Tomcat server ports were open, especially 8080. This can be done using the "netstat" command. For best results, shut down Tomcat and run netstat, just to make sure that some other app hasn't claimed the ports, then try again after starting Tomcat.
A look in the localhost and catalina logs for startup errors is also worth doing. And, by the way, have you made sure that the disks that hold your log and workfiles are not full?
If everything else is clean,
you should try a simple web request such as "http://mytomcathost:8080/fubar.html". If you get a "404 page not found" response, then Tomcat itself is working. At that point, watch things and see if you are degrading into a situation such as the one that Bill has described. Do your tests from external boxes, just to ensure that you don't have firewall problems.
Also, note that I haven't addressed problems that can occur when Apache (or some other server) is serving as a front-end proxy to Tomcat. The rule there is first verify Tomcat itself using the above tests, then check the Apache-to-tomcat connectors.
Some people, when well-known sources tell them that fire will burn them, don't put their hands in the fire.
Some people, being skeptical, will put their hands in the fire, get burned, and learn not to put their hands in the fire.
And some people, believing that they know better than well-known sources, will claim it's a lie, put their hands in the fire, and continue to scream it's a lie even as their hands burn down to charred stumps.