  We formulate quenched chiral perturbation theory for heavy-light mesons
coupled to pions, and calculate the one-loop chiral logarithmic corrections to
$f_B$, $f_{B_{s}}$, $B_B$ and $B_{B_{s}}$. We also calculate these corrections
for ``partially quenched'' theories. In both theories, the chiral logarithms
diverge in the chiral limit, indicating that (partially) quenched theories
should not be used to study this limit. Comparing the chiral logarithms to
those in QCD, we estimate the errors caused by (partial) quenching. By forming
suitable ratios, we can reduce the uncertainties in our estimates.
