Here is an example.

Consider the truncated category $\Delta_{\le n+1}$. Then presheaves on this with the model structure presents $n$-types.

EDIT:
Andrea points out that the relevant Quillen equivalences+proofs can be found in **Corollary 9.2.7** of Cisinski's book _Les préfaisceaux comme modèles des types d'homotopie_.