Here is a classical example.
Let CDGA be the category of commutative differential graded algebras over a fixed ground field k of characteristic p$p$. Weak equivalences are quasi-isomorphisms, fibrations are levelwise surjections. These would determine the others, but cofibrations are essentially generated by maps A → B$A \rightarrow B$ such that on the level of the underlying DGA, B$B$ is a polynomial algebra over A$A$ on a generator x$x$ whose boundary is in A$A$.
CDGA is complete and cocomplete, satisfies the 2$2$-out-of-3$3$ 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$A \rightarrow B$ is a map of commutative DGAs which is a fibration in the above sense. Then for any element [x]$[x]$ in the (co)homology of B$B$ in even degree, the p'th$p$-th power [x]p$[x]^p$ is in the image of the cohomology of A$A$. In fact, pick any representing cycle x in B$x \in B$ and choose a lift y to A$y \in A$. Then the boundary of yp$y^p$ is pyp-1 = 0$py^{p-1} = 0$ by the Leibniz rule, so [yp]$[y^p]$ is a lift of [x]p$[x]^p$ to the (co)homology of A$A$.
(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∞$E_\infty$ DGAs.)
Nothing goes wrong in characteristic zero.
