  A monomial ideal $I$ admits a Betti splitting $I=J+K$ if the Betti numbers of
$I$ can be determined in terms of the Betti numbers of the ideals $J,K$ and $J
\cap K$. Given a monomial ideal $I$, we prove that $I=J+K$ is a Betti splitting
of $I$, provided $J$ and $K$ are componentwise linear, generalizing a result of
Francisco, H\`a and Van Tuyl. If $I$ has a linear resolution, the converse also
holds. We apply this result recursively to the Alexander dual of
vertex-decomposable, shellable and constructible simplicial complexes and to
determine the graded Betti numbers of the defining ideal of three general fat
points in the projective space.
