What is the necessary condition for a sum $Z=X+Y$ of two normal random variables $X$ and $Y$ to be a normal random variable?
If this is too difficult to state in general, what are some sufficient conditions for that (besides the well-known sufficient condition that the random vector $(X,Y)$ is multivariate normal)?
(A related thread Is joint normality a necessary condition for the sum of normal random variables to be normal? exemplifies that joint normality is not a necessary condition.)
