The problem of the prediction of the effective electrical conductivity of a polycrystal from the electrical conductivity of a single crystal is considered. It is shown that the familiar Voigt-Reuss bounds on the behavior of a polycrystal are the very best generally valid bounds that have thus far been proposed and that the various methods that are claimed to predict exact effective conductivity (or narrow bounds) all include implicit restrictions on the internal geometry of the polycrystal. This is accomplished by constructing a series of statistically homogeneous and isotropic polycrystal models for which the effective conductivity can be exactly calculated. It is hence to be expected that no universal relationship between single-crystal and polycrystal conductivity exists. Experimental evidence is adduced to support this conclusion. The results are also applicable to the analogous problems of thermal conductivity and electrical permittivity.