Find independent $X$ and $Y$ that are not normally distributed s.t. $X+Y$ is normally distributed.

According to a theorem conjectured by P. Lévy and proved by H. Cramér (see Feller, Chapter XV.8, Theorem 1),

If $X$ and $Y$ are independent random variables and $X+Y$ is normally distributed, then both $X$ and $Y$ are normally distributed.

I assume that $Y$ being a constant (and hence independent of $X$) is considered as being accounted for in this theorem by considering $Y$ to be a degenerate normal random variable with variance $0$.