What really counts is that you're satisfied with the rationale you've provided. But you've peaked my interest (no pun intended). The spread in the histogram is certainly about what you'd expect from a Poisson with mean around 50: 95% of the counts would fall between 36 and 64. The 60-minute moving average figure looks like what one would expect from 3500 1-minute counts from a Poisson distribution. But samples from a Poisson distribution would not have such a peak in a histogram. The observed histogram is consistent with a random samples from a Poisson distribution but "contaminated" with a set of values very close to the mean.
If you're willing to share one of the datasets (in the original time order) and/or your Mathematica code to get the histograms, I'd certainly like to see if I could see something in the data. (Daniel Lichtblau is absolutely correct: one can't say much without the raw data.) Also, it seems like your rationale could be validated with some additional lines of code.
