  We present models of the components of the systems KOI-126 and CM Draconis,
the two eclipsing binary systems known to date to contain stars with masses low
enough to have fully convective interiors. We are able to model satisfactorily
the system KOI-126, finding consistent solutions for the radii and surface
temperatures of all three components, using a solar-like value of the
mixing-length parameter \alpha in the convection zone, and PHOENIX NextGen 1D
model atmospheres for the surface boundary conditions. Depending on the
chemical composition, we estimate the age of the system to be in the range 3-5
Gyr. For CM Draconis, on the other hand, we cannot reconcile our models with
the observed radii and T_eff using the current metal-poor composition estimate
based on kinematics. Higher metallicities lessen but do not remove the
discrepancy. We then explore the effect of varying the mixing length parameter
\alpha. As previously noted in the literature, a reduced \alpha can be used as
a simple measure of the lower convective efficiency due to rotation and induced
magnetic fields. Our models show a sensitivity to \alpha (for \alpha < 1.0)
sufficient to partially account for the radius discrepancies. It is, however,
impossible to reconcile the models with the observations on the basis of the
effect of the reduced \alpha alone. We therefore suggest that the combined
effects of high metallicity and \alpha reduction could explain the observations
of CM Draconis. For example, increasing the metallicity of the system towards
super-solar values (i.e. Z = 2 Z_sun) yields an agreement within 2 \sigma with
\alpha = 1.0.
