Edited to make this more concrete:
Given a random vector $(X_1,X_2)$ that is jointly normal with means / sd's $\mu_1,\mu_2, \sigma_1,\sigma_2$ and correlation $\rho$, the sum of $S=X_1+X_2$ is normally distributed with mean $\mu_1+ \mu_2$ and variance $\sigma_1^2+\sigma_2^2 + 2\rho \sigma_X \sigma_Y$.
In particular, as $\rho$ increases, the CDF of the sum 'rotates' about $(\mu_1+ \mu_2,1/2).$
Now for other rvs, not normal, intuitively it seems like something like this property should generically be true: when you add two random variables and only their correlation increases (perhaps a more formal correlation order is needed than simply the number $\rho$), the CDF of the sum should rotate about the mean $\mu_X+\mu_Y$.
My intuition is that as the rv's get more correlated the chance that their sum takes on very high or very low values should monotonically decrease. This is clear for the normal case, but can it be formalized for some larger class in general? Any references would be appreciated.