Marginal weak convergence does not control sums

Let $Z$ take values $1,-1$ with equal probabilities. Set $X_n=Z$ for every $n$, and $Y_n=Z$ for even $n$, $Y_n=-Z$ for odd $n$. Both marginal laws are constant, but $X_n+Y_n$ alternates between the law of $2Z$ and a point mass at zero. Thus separate weak convergence of random variables does not determine a joint limit or a sum limit. The obstruction disappears when one marginal limit is constant, as in the Slutsky theorem.