TITLE: Limitations on model-categorical presentations
QUESTION [20 upvotes]: In higher category theory, it is common that a weak structure cannot be strictified in all directions simultaneously.  For instance, a monoidal category is not (in general) equivalent to one that is both strict and skeletal, and a tricategory is not (in general) equivalent to one whose units and interchange law are both strict.
Now a model category can be regarded as a particular sort of strictification of an $(\infty,1)$-category.  From this perspective, all sorts of questions along the above lines suggest themselves.  For concreteness, I'll ask a particular one:

Does there exist a locally presentable $(\infty,1)$-category which (provably) cannot be presented by a model category in which all objects are both fibrant and cofibrant?

But I would be interested in answers to any similar question.

REPLY [3 votes]: Here is another answer that involves adding extra properties.  If we have a model category which 

is locally cartesian closed, as a category (such as if it is a presheaf category)
has its cofibrations being the monomorphisms (hence in particular all objects are cofibrant)
is right proper (such as if all objects are fibrant)

then pullback along a fibration $g\colon A\to B$ preserves both cofibrations and acyclic cofibrations, and so the adjunction $g^* \dashv \Pi_g$ is Quillen.  Therefore, the $(\infty,1)$-category presented by this model category is locally cartesian closed.
Thus, an $(\infty,1)$-category which is not locally cartesian closed cannot be presented by a model category with all three of the above properties.