Well, the poisson distribution is often the first model considered for random counts. To test for it...you should know that the poisson distribution has a property that the mean of distribution is equal to the variance. So if you find out that the variance is substantially larger than the mean, it's not a good approximation. Usually, the negative binomial distribution is considered instead of poisson if the latter fails.
You can get more insights into this theme, for instance, in Rice: Mathematical Statistics and Data analysis.