Maybe that's a very naive question.
Is the subcategory of cofibrant and fibrant objects of a model category a model category itself (with the induced equivalences, cofibrations and fibrations)?
The category of fibrant-cofibrant objects needs not to be complete nor cocomplete.
The other axioms, though, are trivially verified. Even the factorization axiom: if you have an arrow $X \longrightarrow Y$, with $X$ and $Y$ fibrant-cofibrant objects, then with both factorizations $X \longrightarrow Z \longrightarrow Y$ you always get objects $Z$ that are fibrant-cofibrant ones too because, the first arrow being a cofibration and $X$ being cofibrant, implies that $Z$ is also cofibrant. Dually, since the second arrow is always a fibration and $Y$ is fibrant, $Z$ must be also fibrant.