This week's book giveaway is in the General Computing forum. We're giving away four copies of Arduino in Action and have Martin Evans, Joshua Noble, and Jordan Hochenbaum on-line! See this thread for details.
It's nearly 100% certain that the failing application is defective and from the sound of it, it probably has a slow resource leak. Tomcat is very reliable, and when it does fail, it is almost always going to fail for all apps, not just one of them.
Slow resource leaks can take a certain amount of detective work. They often occur because of things like people forgetting to close files (or thinking that they don't have to), database exception handlers that don't close connections, cleanup services that don't actually clean up - causing the number of things being "cleaned" to get bigger and bigger". Stuff like that.
You'll probably need both Tomcat logs and application logs. I would also interview some of the users and see if they start seeing problems (slow response, error pages) that build up the longer the application has been running. That may give you an idea of where to investigate further.
Customer surveys are for companies who didn't pay proper attention to begin with.