I am going to make this question deliberately vague.

A category is concrete if its objects can be realized as sets with extra structure (in particular, it admits a faithful functor to Set).

The examples I know of non-concretizable categories (the homotopy category, Cat with naturally isomorphic functors identified) are obviously derived from forgetting structure of higher categories. Is there some sense in which, given any locally small category, the objects of that category can be realized themselves as (small) higher categories with extra structure, analogous to considering faithful functors to Set?

EDIT: I think my question is as Jeremy suggested below: Given a locally small category, is it always the homotopy category of a concrete $(\infty,1)$-category, for some reasonable definition of concrete?