  The results of a multi-lepton search conducted by the CMS collaboration with
5 fb$^{-1}$ of data collected from 7 TeV $pp$ collisions are used to place the
first bound on the rare flavor-changing decay of the top quark to a Higgs boson
and charm quark. Combining results from a number of exclusive three- and
four-lepton search channels yields an estimated upper limit of ${\rm Br}(t \to
ch) < 2.7%$ for a Higgs boson mass of 125 GeV. The sensitivity of future
dedicated searches for $t \to c h$ could be improved by adding exclusive same
sign di-lepton channels, as well as by sub-dividing channels based on $b$-quark
tagging and partial kinematic top quark and Higgs boson tagging. This bound may
be interpreted more widely within a range of new physics processes that yield
final states with a $W$-boson in association with a Higgs boson. For such
processes with kinematics that are similar to top--anti-top production and
decay, the estimated limit on cross section times branching ratio corresponds
to roughly $\sigma \cdot {\rm Br}(pp \to WhX) < 9$ pb.
