You can just consider a symmetric rv : Xn such that Xn has the same distribution as -Xn.
So we can consider Xn and Yn=-Xn, which is not independent of Xn. It is true to say that Xn converges to a rv X and Yn converges to a rv Y that follows the same distribution as X.
But Xn+Yn=0 doesn't converge to X+Y.
It converges weakly to X, such that .
Same for and Y.