Are $X$ and $Y$ necessarily normal if the the sum $Z=X+Y$ is normal?

Question

Of course, asking the question the other way round is straightforward to answer as via the convolution we find that the sum of two normal distributed variables is again normal.

But however, is it possible that $X$ and $Y$ follow different distributions but the sum is again normal?

Answer 1

If X and Y are independent, then the result follows from Cramer's Theorem.

Here is a link. Note the requirement of independence here.

In the non-independence case, if $Y=Z-X$, where $Z$ is normal, and $X$ has any distribution also works.