NP-complete problem with a polynomial number of yes-instances?
A language with only polynomially many yes-instances is called sparse.  Mahaney's theorem states that if any NP-complete language is sparse, then P = NP.  Since most people expect that P $\ne$ NP, it seems unlikely that there exists a NP-complete language with only polynomially many yes-instances.

It's a separate question whether the number of yes-instances is exponential.  (One could imagine that the number of yes-instances might be more than polynomial but less exponential.)  The Berman-Hartmanis conjecture is relevant here; it implies that all NP-complete problems have exponentially many yes-instances.  The conjecture remains an open problem.