The condition you're looking for is called combinatoriality (and local presentability). A model category is combinatorial provided it satisfies some complicated conditions involving accessibility, but at least when the underlying category is a presheaf topos and the cofibrations are exactly the monomorphisms, we can actually systematically construct these model structures using the framework of Denis-Charles Cisinski in his book Les Préfaisceaux comme modèles des types d'homotopie (Astérisque 308). This, in particular gives us the combinatoriality of the model structure on simplicial sets essentially for free.
For a general reference on combinatoriality and Jeff Smith's theorem, I suggest you take a look at the papers Sheafifiable Homotopy Model Categories by Tibor Beke and On Left and Right Model Categories and Left and Right Bousfield Localizations by Clark Barwick. (The original theorem is due to Jeff Smith, but he has neglected to publish it (it was announced at a conference and the main ideas were given, I believe), although he has assured us many times that he is in the process of publishing a book).