We have four random variables say W,X,Y,Z where W and X has the same distribution and Y, Z also has 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 where defined we used linearity and it is obvious, also 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 $E(Z+W)=0$.