The error message I see says the maximum connections are exceeded, not the maximum SQL connections. It's an ASP error that is "supposed" to mean that more people are trying to use the site than the maximum you've configured it to allow.
At least that's what Microsoft says the error means.
FWIW, the "request queue" they mention that you should watch to see if requests are getting backed up is constantly zero until ASP dies. After ASP dies, the queued requests start stacking up. Even with about 1.1 million pageviews per day (about 2.5 million raw IIS requests), everything gets processed so quickly the queue rarely even gets ONE entry put in it.
Edit: Upgrading to Windoze 2003/IIS6 likely won't solve the problem but it looks like it could give me some handy tools to help try to nail it down. And possibly allow me to configure it to automatically restart ASP anytime it's shut down.