Techniques for proving NP completeness for a specific sequence of instances
If I understand the question correctly, I think that you are essentially asking whether a problem can be NP-hard even when it is defined on a sparse set of input strings.  If your notion of NP-hardness is under polynomial-time many-one reducibility, such a problem cannot exist (unless P=NP) because of Mahaney’s theorem.

See a blog article about Mahaney’s theorem by Lance Fortnow for a nice explanation of Mahaney’s theorem and related results.