I found a quicker way to get the counter to go up, and that was to just paste the address to the counter into Firefox and refresh just that.
Since it looks like a PHP counter, you don't even have to wait for the server to respond to your request-- just click refresh, wait until the browser
tells you the server received your request and is awaitng a response, click refresh again. I was able to do this fast and furiously... I hope that wasn't what killed it...
EDIT:
Found this on the hit counter provider's website FAQ:
My counter suddenly stopped showing up. What is wrong?
As mentioned above, our counter is good for most personal and business sites with reasonable traffic. However if you are pounding your free counter real hard, our system may temporarily stop showing your counter. If you find that your counter has stopped working permanently, then we probably have banned your account for violation of our Terms and Conditions.