We have four random variables say $W,X,Y,Z$ where $W$ and $X$ have the same distribution and $Y$, $Z$ also have the same distribution. Bad news is $EX$ and $EY$ may not exist but $E(W+Z)$ is zero. If we assume that $E(X+Y)$ exists, could we conclude that $E(X+Y)$ is zero?
I know if $EX$ and $EY$ were defined, we used linearity and it is obvious. However, we know nothing more about $X$, $Y$.
Correlation may kill the cat.
If the distribution is (forsimplicity) symmetric about the origin, we may have that $Z=-W$ whereas $X,Y$ are independent, even though all four variables have the same distribution. With $X,Y$ independent $E(X+Y)$ will not exist if $E(X),E(Y)$ don't exist. But $Z+W=0$ a.s., hence $E(Z+W)=0$.