Consider two random variables, $X \sim N(\mu_X, \sigma_X^2)$ and $Y \sim (\mu_Y, \sigma_Y^2)$. It is known that if $X$ and $Y$ are independent, then the sum/difference of $X$ and $Y$ will be normally distributed, such that $X \pm Y \sim N(\mu_X \pm \mu_Y, \sigma^2_X+\sigma^2_Y)$.
However, is the converse true? Supposing we have $2$ independent random variables $X$ and $Y$ such that $X \pm Y $ is normally distributed, is it then true that $X$ and $Y$ must both be normally distributed? In other words, is it possible for the sum/difference of $2$ non-normally distributed random variables to be normally distributed?