  In order to address this question, we consider a simple renormalisable and
gauge invariant model in which the $Z'$ only has couplings to the electron and
muon and their associated neutrinos, arising from mixing with a heavy
vector-like fourth family of leptons. Within this model we discuss the
contributions to the electron and muon anomalous magnetic moments from $Z'$
exchange, subject to the constraints from $\mu \rightarrow e \gamma$ and
neutrino trident production. Using analytic and numerical arguments, we find
that such a $Z'$ model can account for either the electron or the muon $g-2$
anomalies, but not both, while remaining consistent with the experimental
constraints from $\mu \rightarrow e \gamma$ and neutrino trident production.
