  We introduce a model for Ce$^{3+}$ impurities in cubic metals which exhibits
competition between the Fermi-liquid fixed point of the single channel Kondo
model and the non-Fermi-liquid fixed point of the two- and three-channel Kondo
models. Using the non-crossing approximation and scaling theory, we find: (i) A
possible three-channel Kondo effect between the one- and two-channel regimes in
parameter space. (ii) The sign of the thermopower is a fixed point diagnostic.
(iii) Our results will likely survive the introduction of additional $f^2$ and
conduction states. We apply this model to interpret the non-Fermi liquid alloy
La$_{1-x}$Ce$_x$Cu$_{2.2}$Si$_2$.
