Is it true that $I(X;Y)+I(Y;Z)=I(X;Z)$ for $X \to Y \to Z$?
$I(X;Z) = H(X)+H(Z)-H(X,Z)$ and $I(X;Y)+I(Y;Z) = H(X)+H(Z)-H(Z|Y)-H(X|Y)$
Hence, we would require $-H(X,Z)=-H(Z|Y)-H(X|Y)$ -- is it true?
Also, $I(X;Y,Z) = I(X;Y)+I(X;Z|Y) = I(X;Y)$ due to conditional independence. Then $I(X;Y)+I(Y;Z) = I(X;Y,Z)+I(Y;Z)$, which doesn't seem to help.
It is definitely wrong, simply because of DPI. You can check that $I(X;Y)=I(X;Z)+I(X;Y|Z)$.