SAT Problems have a phase transition that depends on the ratio $r$ of variables to clauses. Below $r$, SAT problems are solvable quickly; above, they become difficult. The same is true of NP Complete problems in general, AFAIK.
Suppose I have a hard instance of an NP Hard problem, say HAM CYCLE and I reduce the problem down to SAT. Can you say anything about the hardness of the SAT instance? Will it be easy or hard?
Intuitively, it seems the SAT Problem reduced to should be hard, but is there a proof?