Backdoors and backbones of Boolean formulas are hidden structural properties. A natural goal, already in part realized, is that solver algorithms seek to obtain substantially better performance by exploiting these structures. However, the present paper is not intended to improve the performance of SAT solvers, but rather is a cautionary paper. In particular, the theme of this paper is that there is a potential chasm between the existence of such structures in the Boolean formula and being able to effectively exploit them. This does not mean that these structures are not useful to solvers. It does mean that one must be very careful not to assume that it is computationally easy to go from the existence of a structure to being able to get one's hands on it and/or being able to exploit the structure. For example, in this paper we show that, under the assumption that P $\neq$ NP, there are easily recognizable families of Boolean formulas with strong backdoors that are easy to find, yet for which it is hard (in fact, NP-complete) to determine whether the formulas are satisfiable. We also show that, also under the assumption P $\neq$ NP, there are easily recognizable sets of Boolean formulas for which it is hard (in fact, NP-complete) to determine whether they have a large backbone.