In reply to Max.

Those are interesting observations. Thanks for your thoughts and experiments. I've verified that Friendly's %POIPLOT macro produces results that are similar to my implementation, so the "problem" does not seem to be my implementation.

I agree with you that the Poissonness plot *by itself* might not adequately indicate the quality of the fit. Here are a few other thoughts:
A) Diagnostic plots such as this and Q-Q plots are visual guides, not definitive tests.
B) If you make a normal Q-Q plot from truly normal data by using RAND("Normal"), you will see the same things with respect to a straight line in the middle but "curves" in the tails. That is the result of sample variation.
C) In the previous article, I proposed three steps to fitting a Poisson model to data: A GENMOD model, a density overlay on a bar chart, and a Q-Q or Poissonness plot. When you do steps 1 and 2 it is clear that a Poisson model does not fit your data with the extra 13s.

The conclusion I draw from your investigation is that we shouldn't use the Poissonness plot in isolation. Like all tests, it can give false positives and false negatives.