I have had this problem in the past, and it is due to too much liquid. What happens is that the fruit exudes even more liquid when it bakes, adding to the liquid already in the pie pan, and it is like a rising tide that submerges the lattice. Sometimes I have had to use a turkey baster in the middle of baking to suck out some of the juice to keep a total, overflowing flood from happening.
My current solution is to pre-cook the filling before I put it in the pie. If you look at one of the cherry pie recipes in PPB (I think it is the Designer Cherry Pie), Rose gives directions for pre-cooking the filling, and then putting it into the unbaked crust. I have done this on the last few cherry pies I’ve made, and it has worked very well. I have also used Pomona’s Universal Pectin to firm the filling (as opposed to using cornstarch), because I simply prefer the texture.
I have also begun to pre-cook my apples for double-crust apple pies. I like the slight carmelized flavor of apples sauteed in butter (which is what you would do for a tarte tatin), and I find it works well for a classic American apple pie. I simply put the cooled filling into the unbaked shell and freeze the pie until I am ready to finish baking it. The advantage of a pre-cooked filling is that you never have to pay attention to when it is cooked. You need only watch to be sure the crust is done.