One of the classic examples of a fluid mechanical bifurcation is the appearance of Taylor vortex flow in a viscous fluid contained between concentric rotating cylinders. A natural consequence of the mathematical model due to D. G. Schaeffer, which has been most successful in describing this phenomenon in a finite system, is the presence of `anomalous modes' in the steady solution set. These are cellular flows that exist in contradiction to the commonly held belief that the radial flow close to the ends is necessarily directed towards the inner cylinder because of the presence of an Ekman boundary layer. To date anomalous modes have only been studied in a restricted parameter range, but here we report the results of numerical and experimental studies that cover a wide range of parameter space. It is shown that the lower stability limits for anomalous modes are removed to high Reynolds numbers when the gap between the cylinders is reduced. This perhaps explains why this class of flow remained undiscovered in over fifty years of study of the Taylor-Couette problem until the work of T. B. Benjamin. We also consider the implications of these results for the organization of dynamical behaviour in the Taylor-Couette system.