Is Poisson a Good Use Case?

If we're trying to determine whether a particular set of behaviors or parameters is a good candidate for the Poisson distribution, again, it doesn't fit that criteria. The occurrences of the event are independent of each other. And the likelihood of the event occurring at any given point in time is more or less uniform. Is there anything else that we can use as kind of a shortcut for determining if this is a good or a bad use case for Poisson?