True. In the case that X=aY+Z, it does not hold also if Corr(Y,Z) is different from zero and/or E(Z) is different from zero (I forgot to mention this condition before).
In a paper, I have found this relationship:
E(X|Y<y)=Cov(X,Y)*E(Y|Y<y)/Var(Y)
It seems to me that the previous equation holds if, for instance, X=aY+Z with Z and Y independent and a non zero.
It also holds if (X,Y) is a bivariate normal (with non zero correlation).
But does it hold...