This theorem is stated as a footnote in a lecture:
Let $X,Y$ be independent random variables
If $X+Y$ follows a normal distribution, then $X$ and $Y$ follow a normal distribution.
If $X+Y$ follows a Poisson distribution, then $X$ and $Y$ follow a Poisson distribution.
Note that $X$ and $Y$ are not supposed to be identically distributed. $X$ may follow Poisson(1) while $Y$ follows Poisson(2).
I've been trying to prove these statements without success. Sums of independent random variables hint at something involving characteristic functions (inverse Fourier transform or some such), but I've been unsuccessful so far.