Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let G be a kG-connected graph and D_c(G) denote the diameter of G after deleting any of its c \lt kG vertices. For a product of three factors G_1, G_2 and G_3, we prove that D_a+b+c+2(G) \lt D_a(G_1) + D_b(G_2) + D_c(G_3) + 1. We indicate how analogous proof gives the upper bound D_a+b+1(G) \lt D_a(G_1) + D_b(G_2) + 1 for the product of two factors. Finally, we show that D_a+b+1(G) \lt D_a(F) + D_b(B)+1 if G is a graph bundle with fibre F over base B, a \lt k_F,and b \lt k_B.