The performance of characteristic methods (CMs) on problems that contain optically thick diffusive regions is analyzed and tested. The asymptotic analysis holds for moment-based characteristics methods that are algebraically linear; for one-, two-, and three-dimensional Cartesian coordinate systems; and for arbitrary spatial grids composed of polygons (two dimensions) or polyhedra (three dimensions). The analysis produces a theory that predicts and explains how CMs behave when applied to thick diffusive problems. The theory predicts that as spatial cells become optically thick and highly scattering, CMs behave almost exactly like discontinuous finite element methods (DFEMs). This means that there are two classes of CMs: those that fail dramatically on thick diffusive problems and those whose solutions satisfy discretizations of the correct diffusion equation. Most CMs in the latter set behave poorly in general, sometimes producing oscillatory and negative solutions in thick diffusive regions. However, the analysis suggests that certain reduced-order CMs, which use less information on cell surfaces than is readily available, will behave more robustly in thick diffusive regions. The predictions regarding standard CMs are tested by using the linear and bilinear characteristics methods on several test problems with rectangular grids in x-y geometry. The predictions regarding reduced-order CMs are tested by solving x-y test problems on triangular grids using a CM that employs linear functions for cell-interior sources but constants for cell-surface fluxes. In every case the numerical results agree precisely with the predictions of the theory.