  We prove existence results for small presentations of model categories
generalizing a theorem of D. Dugger from combinatorial model categories to more
general model categories. Some of these results are shown under the assumption
of Vop\v{e}nka's principle. Our main theorem applies in particular to
cofibrantly generated model categories where the domains of the generating
cofibrations satisfy a slightly stronger smallness condition. As a consequence,
assuming Vop\v{e}nka's principle, such a cofibrantly generated model category
is Quillen equivalent to a combinatorial model category. Moreover, if there are
generating sets which consist of presentable objects, then the same conclusion
holds without the assumption of Vop\v{e}nka's principle. We also correct a
mistake from previous work that made similar claims.
