Can I bound the cardinality of a set if testing for membership in it is known to be NP-complete?
Mahaney’s Theorem states that sparse NP-complete sets exist iff P=NP. Therefore, assuming $P \neq NP $ implies a super-polynomial lower bound on the number of instances of size $n$ in $NP$-complete sets, for infinitely many $n$. That is, if $P \neq NP$, then any $NP$-complete set must have some $\epsilon \gt 0$ such that for infinitely-many integers $n\ge 0$, the set contains 
at least $2^{n^{\epsilon}}$ strings of  length $n$. 

H. Buhrman and J. M. Hitchcock proved the lower bound ($2^{n^{\epsilon}}$) is tight, unless the polynomial hierarchy collapses. 

[1] H. Buhrman and J. M. Hitchcock, NP-Hard Sets are Exponentially Dense Unless coNP ⊆ NP/poly, In IEEE
Conference on Computational Complexity, pages 1–7, 2008

[2] Eric Allender, A Status Report on the P Versus NP Question, Advances in Computers, Volume 77, 2009, Pages 117-147