Read off from second fundamental form if submanifold is contained in a larger totally geodesic submanifold?

Question

Let $Z$ be a Riemannian manifold and consider submanifolds $X\subset Y\subset Z$. Denote by $N_{X,Y}$ the normal bundle of $X$ in $Y$ and $N_{X,Z}$ of $X$ in $Z$ and $N_{Y,Z}$ of $Y$ in $Z$. Let $\Pi_{X,Y},\Pi_{X,Z},\Pi_{Y,Z}$ be the respective second fundamental forms. When pulling back/restricting the bundles to $X$ the metric on $Z$ gives a splitting $N_{X,Z}=N_{X,Y} \oplus N_{Y,Z}$. With this identification, I have convinced myself that $\Pi_{X,Z}=\Pi_{X,Y}+\Pi_{Y,Z}$. In other words, if $Y$ is totally geodesic then $\Pi_{X,Z}$ takes values in a proper subbundle of $N_{X,Z}$. My question is: Is the converse also true? Ie. if $\Pi_{X,Z}$ takes values in a proper subbundle, is there a totally geodesic submanifold $Y\subset Z$ such that $X\subset Y$. I suppose all we can hope for is a local statement but maybe more can be said when some of the submanifolds are compact and simply-connected or something like that?