[Over-dispersion][1] can occur with one-parameter distributions, where mean and variance are tied together (Poisson, Binomial, Exponential). In real data, variance is usually much greater than would be allowed. Over-dispersion creates over-confidence, but usually does not introduce biases. In practical modelling, this problem  can be resolved in one of two ways: using two-parameter distributions or observation-level random effects.

[1]: https://schmettow.github.io/New_Stats/glm.html#overdispersion