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 $A \rightarrow B$ 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 $A \rightarrow B$ 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 $[x]^p$ is in the image of the cohomology of $A$.  In fact, pick any representing cycle $x \in B$ and choose a lift $y \in A$.  Then the boundary of $y^p$ is $py^{p-1} = 0$ by the Leibniz rule, so $[y^p]$ is a lift of $[x]^p$ 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 $E_\infty$ DGAs.)

Nothing goes wrong in characteristic zero.