  We investigate possible generalizations of the Coffman-Kundu-Wootters
monogamy inequality to four qubits, accounting for multipartite entanglement in
addition to the bipartite terms. We show that the most natural extension of the
inequality does not hold in general, and we describe the violations of this
inequality in detail. We investigate alternative ways to extend the monogamy
inequality to express a constraint on entanglement sharing valid for all
four-qubit states, and perform an extensive numerical analysis of randomly
generated four-qubit states to explore the properties of such extensions.
