  A general method is proposed for predicting the asymptotic percolation
threshold of networks with bottlenecks, in the limit that the sub-net mesh size
goes to zero. The validity of this method is tested for bond percolation on
filled checkerboard and "stack-of-triangle" lattices. Thresholds for the
checkerboard lattices of different mesh sizes are estimated using the gradient
percolation method, while for the triangular system they are found exactly
using the triangle-triangle transformation. The values of the thresholds
approach the asymptotic values of 0.64222 and 0.53993 respectively as the mesh
is made finer, consistent with a direct determination based upon the predicted
critical corner-connection probability.
