  We consider the 2D quenched--disordered $q$--state Potts ferromagnets and
show that at self--dual points any amalgamation of $q-1$ species will fail to
percolate despite an overall (high) density of $1-q^{-1}$. Further, in the
dilute bond version of these systems, if the system is just above threshold,
then throughout the low temperature phase there is percolation of a single
species despite a correspondingly small density. Finally, we demonstrate both
phenomena in a single model by considering a ``perturbation'' of the dilute
model that has a self--dual point. We also demonstrate that these phenomena
occur, by a similar mechanism, in a simple coloring model invented by O.
H\"aggstr\"om.
