Here is a classical example. Let CDGA be the category of commutative differential graded algebras over a fixed ground field k of characteristic p. Weak equivalences are quasi-isomorphisms, fibrations are levelwise surjections. These would determine the others, but cofibrations are essentially generated by maps A → B such that on the level of the underlying DGA, B is a polynomial algebra over A on a generator x whose boundary is in A. CDGA is complete and cocomplete, satisfies the 2-out-of-3 axiom, the retract axiom, satisfies lifting, and a general map can be factored into a cofibration followed by an acyclic fibration by the small object argument. However, you don't have factorizations into acyclic cofibrations followed by fibrations, because of the following. Suppose A → B is a map of commutative DGAs which is a fibration in the above sense. Then for any element [x] in the (co)homology of B in even degree, the p'th power [x]<sup>p</sup> is in the image of the cohomology of A. In fact, pick any representing cycle x in B and choose a lift y to A. Then the boundary of y<sup>p</sup> is py<sup>p-1</sup> = 0 by the Leibniz rule, so [y<sup>p</sup>] is a lift of [x]<sup>p</sup> to the (co)homology of A. (As a result, there are a lot of other "homotopical" constructions, such as homotopy pullbacks, that are forced to throw you out of the category of commutative DGAs into the category of E<sub>∞</sub> DGAs.) Nothing goes wrong in characteristic zero.