Prepare Pastry: In a large bowl, combine flour and salt. Using a pastry blender or large fork, cut in the butter until pea-size pieces remain. Add shortening and cut in until mixture resembles coarse crumbs. With a fork, gradually stir in the water until combined. Gather dough together and divide into two balls. Flatten each ball to a 6-inch disk; wrap the dough in plastic and refrigerate at least 1 hour.
Make Filling: In a large bowl, combine blueberries, cherries, sugar, and tapioca. Mix well; let stand 15 minutes.
Heat oven to 400 degrees F. To assemble pie: On lightly floured surface, roll out one piece of dough into an 11 1/2-inch circle; fit into a 10-inch pie pan. Roll out remaining dough into an 11-inch circle; cut into 12 strips, each about 3/4-inch wide; discard trimmings. Fill lined pie pan with fruit mixture. Weave strips into a lattice pattern over fruit. Trim edges; pinch top and bottom crusts together.
Bake pie 20 minutes; reduce heat to 375 degrees F and bake until crust is golden and filling is bubbly, 30 to 40 minutes more. (If necessary, cover edges of crust with aluminum foil to prevent over-browning.) Let pie cool completely before serving.
