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 (not me specifically!) many times that he is in the process of publishing a book).