Hermite Distribution: Is Wikipedia wrong? While clicking around in Wikipedia, I ended up at a page for Hermite Distribution, apparently a useful 2-parameter substitute for the 1-parameter Poisson distribution; for example "McKendrick considered the distribution of counts of bacteria in leucocytes [and found the] Hermite distribution ... more satisfactory than fitting ... with a Poisson distribution."
But look at the graph at the top of the Wikipedia page!  Do you really believe that the bacteria counts are much more likely to be odd numbers than even numbers?  I'll guess that the correct Hermite distribution is derived from an expression like floor(f_1 + 2·f_2) but Wiki substituted floor(f_1) + 2·floor(f_2).  Googling around, I found the same "error" — if it is an error — and indeed the same graph in a scholarly paper.
 A: An obvious way for such a distribution to arise is whenever you can observe values occuring in pairs as well as singly; if you regard both the singles- and pairs-processes as independent Poisson processes you'll have a total count of the form W+2Z, where W and Z are independent Poisson. If the mean of W is small compared to Z, you'll see clear alternation in probabilities in that total count. 
Another way - as explained at the Wikipedia article you link - the Hermite can arise is as the distribution of the sum of two correlated Poissons.
I'll describe a simple bivariate model that fits with that comment (though apparently not the only one; it looks like there's other ways to get the Hermite distribution).
Let there be some underlying variable, Z, which is Poisson. We don't observe Z.
Let there be two observed quantities, X, and Y, both Poisson, which arise as X=U+Z and Y=V+Z for U, V and Z all independent Poisson. (We don't observe U or V either.)
Clearly X and Y are correlated with Poisson margins; the bivariate distribution would reasonably be regarded as a bivariate Poisson distribution (indeed it's perhaps even the most commonly used one). 
Here's a sample from that bivariate distribution for a particular set of parameter choices. I have jittered the values a little so it's easier to see them:

The population mean of X is 9 and that of Y is 12; the sample means are quite close to those (the mean of Z is 7 here but we won't know that in practice).
It seems like a fairly plausible simple model for dependent Poissons.
Now what happens if we're interested in the sum, X+Y?
Let W=U+V; W is Poisson and independent of Z.
Then X+Y = W + 2Z ... which is exactly what we have with the Hermite.
That is, if you're in a bivariate situation like that one described above, when looking at the sum, you actually will expect to see the Hermite arise.
For some specific parameter choices, yes, you really would see peaks on alternating values. This would tend to happen if the correlation was high and the mean of W was small (correspondingly, you'd expect to see it if the variance of X-Y was small and their correlation was high; note that X-Y would be Skellam). 
In a case like the one above, though, it just looks like a fairly smooth discrete distribution:

Here's a bivariate example which produces alternating peaks and dips in the distribution of the sum:

It's a bit hard to see there but most of the mass is on the diagonal.
This occurs because X+Y only differs a little from 2Z (W is small). In this case the peaks are at even values, not odd ones; adding W is mostly adding 0, much less often adding 1 (and much more rarely anything else), so mostly you're getting even values.
I think there could be a small mistake in the plot on the Wikipedia page (in the labelling on the x-axis), in that it's showing peaks at odd values rather than even values, but the alternating-peaks effect when the correlation is extremely high is not an error. My second example should be the same as the blue points in the pmf plot on the Wikipedia page, and the sample density of X+Y does look like it if you subtract 1 from the values on the x-axis on the Wikipedia plot.
As to the question of whether this alternating effect is really seen in practice. McKendrick (1926) -- see the Wikipedia page you linked for a full reference  -- does describe realistic experiments where indeed you could expect to see such alternating peaks, e.g. one where leukocytes ingest bacteria either singly (W) or in pairs (Z) - see example 4 p 110 - and gives data which clearly have that alternating feature. I don't know whether the data in that example are real though his text suggests they are. This would match the first situation I described.
He also describes spread of infections that arise from within the same house ('internal') or from external exposure but where the internal rate of infection is much greater than the external (producing the "high correlation" feature I described before); in that case the count of infections within houses could in some cases show that peak alternation (though you wouldn't necessarily expect them to arise in practice).
