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.