For the first time, I decided to carefully maintain the final exams in the order that they were completed, and see if this was at all correlated with the score. This is a 101 course for non-majors and majors with limited prior experience. There is a somewhat reasonable correlation (0.56), although it gets pretty noisy in the middle. I used the rank instead of the raw score (which actually would give -0.6). Blue diamonds are the data points and the gray plus signs are the linear regression (didn't feel like figuring out how to mix scatter plot and line graph!)
Note this isn't measuring the amount of time to complete the exam. The lowest grade was also the last to finish, but s/he showed up extremely late and spent a lot less time on the exam than others. Tendency to show up late is also of course correlated with poor performance. :)