I did all the calculations in December 2003 (you have all the information you need by then). By January 3rd I had all that information typed into the forms ready to be sent.
However, I waited until April 13th to actually write the check and send it in. To me, that is basically the last minute. Why you ask? Since I wanted to use that money for other purposes for those 4 months. This little interest it earned me is a nice (small) bonus as well. I ask you this: why would I want to mail in a check 4 months before it is due?
If you respond that you do it early to get a refund sooner, then I have a second question for you. Why would I pay more than I need to throughout 2003 just to get a return (ie it is better to owe a little bit on your taxes than get a return)?