This paper presents a thorough analysis of the computational performance of a coupled cubic Hermite boundary element/finite element procedure. This C1 (i.e., value and derivative continous) method has been developed specifically for electropotential problems, and has been previously applied to torso and skull problems. Here, the behavior of this new procedure is quantified by solving a number of dipole in spheres problems. A detailed set of results generated with a wide range of the various input parameters (such as dipole orientation, location, conductivity, and solution method used in each spherical shell [either finite element or boundary elements]) is presented. The new cubic Hermite boundary element procedure shows significantly better accuracy and convergence properties and a significant reduction in CPU time than a traditional boundary element procedure which uses linear or constant elements. Results using the high-order method are also compared with other computational methods which have had quantitative results published for electropotential problems. In all cases, the high-order method offered a significant improvement in computational efficiency by increasing the solution accuracy for the same, or fewer, solution degrees of freedom.