  We study the simplest class of Bekenstein-type, varying $\alpha$ models, in
which the two available free functions (potential and gauge kinetic function)
are Taylor-expanded up to linear order. Any realistic model of this type
reduces to a model in this class for a certain time interval around the present
day. Nevertheless, we show that no such model is consistent with all existing
observational results. We discuss possible implications of these findings, and
in particular clarify the ambiguous statement (often found in the literature)
that ``the Webb results are inconsistent with Oklo''.
