This paper attempts to address the question of why certain dataflow analysis problems can be solved efficiently, but not others. We focus on flow-sensitive analyses, and give a simple and general result that shows that analyses that require the use of relational attributes for precision must be PSPACE-hard in general. We then show that if the language constructs are slightly strengthened to allow a computation to maintain a very limited summary of what happens along an execution path, inter-procedural analyses become EXPTIME-hard. We discuss applications of our results to a variety of analyses discussed in the literature. Our work elucidates the reasons behind the complexity results given by a number of authors, improves on a number of such complexity results, and exposes conceptual commonalities underlying such results that are not readily apparent otherwise.