  We study the Artinian reduction $A$ of a configuration of points $X \subset
{\mathbb P}^n $, and the relation of the geometry of $X$ to Lefschetz
properties of $A$. Migliore initiated the study of this connection, with a
particular focus on the Hilbert function of $A$, and further results appear in
work of Migliore--Mir\'o-Roig--Nagel. Our specific focus is on Betti tables
rather than Hilbert functions, and we prove that a certain type of Betti table
forces the failure of the Weak Lefschetz Property (WLP). The corresponding
Artinian algebras are typically not level, and the failure of WLP in these
cases is not detected in terms of the Hilbert function.
