This table is yet unfinished.
If I observe the sum of two processes with known distributions, the distribution of the observations is expected to be …
normal if many summands
|Binomial||Binomial, if common p,
Poisson, if many summands,
Poisson binomial otherwise, note also the binomial sum variance inequality and this.
Wikipedia has some more. A general discussion of probability distribution convolutions is for example here.