Background:
I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This refers to the fact that they can be glued.
For instance, suppose that $X$ is a complex analytic space and $P_1^{\bullet}, P_2^{\bullet}$ are perverse sheaves defined on the open sets $U_1, U_2$ respectively. Then if there exists an isomorphism $\alpha_{ij}: P_1^{\bullet}|_{U_1 \cap U_2} = P_2^{\bullet}|_{U_1 \cap U_2},$ then there exists a unique (up to canonical isomorphism) perverse sheaf $P^{\bullet}$ defined on $U_1 \cup U_2$ such that $P|_{U_i}$ is isomorphic to $P_i^{\bullet}.$
More generally, if one has an open cover $\{U_i\}$ of $X$ and perverse sheaves $P_i^{\bullet}$ of with isomorphisms on the overlaps $U_i \cap U_j$ satisfying the co-cycle condition, then this data glue in the usual way.
My question:
What goes wrong if one tries to glue ordinary complexes of sheaves? Are there counterexamples showing that the gluing property cannot hold in $C^{\bullet}(Sh(X))$ (the category of sheaves on $X$) or in $D^b(X)$ (the bounded derived category of sheaves on $X$)?