Is it possible for the sum of two independent random variables (where at least one of them is not normal) to sum to a normal random variable? Let $X$ be a normal random variable. Suppose we have the following decomposition:
$$
X = Y + Z
$$
where $Y$ ad $Z$ are independent. Is it possible for either $Y$ or $Z$ to be not normal?
I suspect yes but am having trouble coming up with a counter example.

No, it is not possible.
It is a famous result of Cramér that if the sum of two independent random variables $X + Y$ is a normal random variable, then $X$ and $Y$ are normally distributed as well.
This is a difficult result whose proof uses the machinery of complex analysis. The original paper can be found here and the wikipedia page for this result here.
