Here are three examples where poisson statistics are wrong or slightly-wrong for a PMT (photomultiplier tube) in photon-counting mode. These are rather unusual cases -- poisson statistics occur almost always -- but maybe they will help you better understand how poisson statistics come about.
(1) A fancy apparatus deterministically emits exactly one photon per second, and the PMT captures every one of those photons.
Explanation: When there are a fixed number of photons, and almost all of those photons successfully reach and trigger the PMT, then the fact that one photon was measured makes it less likely that there are other photons out there to be captured. This is an extremely unusual case: By contrast, you can imagine, say, a PMT capturing light from a faraway star. The star is emitting grillions of photons; the fact that one flew into your PMT neither raises nor lowers the probability that any other photon from the star will fly into your PMT. In general, Poisson statistics appear when each event does not affect the probability of occurrence of other events.
(2) Every second, my laser fires, and then PMT receives a burst of photons arriving almost-simultaneously (within a picosecond).
Explanation: The PMT needs some recovery time between photons to register them as separate events. There is almost definitely a poisson distribution of how many photons reach the PMT, but there will NOT be poisson distribution in how many photon-counts are registered ... there can only be zero or one current pulse within a picosecond. (This fact is often ignored because under many circumstances, there is negligible chance that two photons arrive so close together. But in pulsed-laser experiments it's often important.) Another way of thinking about this is: The fact that one pulse is measured decreases (to zero) the probability that another pulse will be measured, because of the PMT's dead time. Again, Poisson statistics appear when each event does not affect the probability of occurrence of other events.
(3) The PMT photon-counting threshold is set too high, so even if a photon arrives, it only has a (say) 30% chance of triggering a count.
Explanation: Well, there WILL be a poisson-distribution of photon counts, but the mean number of photon-counts will be only 30% of the mean number of photons. This is sort of a stupid example, sorry.