A 4-point measurement of an insulator would show zero current, as far as I know.

A more plausible idea IMO is that it could be a 4-point measurement of a highly inhomogeneous sample. If the current probes make contact with conductive parts of the sample, and there is a conductive path between them, but one or both voltage probes are not contacting the conductive path, then the voltage measurement would be garbage and potentially zero.

I don’t think this is much of a problem when measuring conventional conductors (you don’t need a particularly good contact, which is much of the point), but a lot of superconductors are oddball materials where, under conditions where they fail to superconduct, they barely conduct at all.

So maybe the sample has a conducting phase and an insulating phase?