Examples of Brown (co)fibration categories that are not Quillen model categories? K.S. Brown has shown that much of abstract homotopy theory can be carried out in the setting of Brown (co)fibration categories [MR0341469]. The decisive property, immediate from the axioms, is that any morphism can be factored into a cofibration, followed by a weak equivalence.
H.-J. Baues [MR0985099], Cisinski [MR2729017] and Radulescu-Banu [arxiv.org/abs/math/0610009] then followed a similar path. Now someone wants to convince me that this is the proper setting for abstract homotopy theory. To begin with, I do like the simplicity of the axioms. Still, I'd like to be convinced of the practical necessity of this approach. Therefore my question:
Are there examples of Brown (co)fibration categories that are not already Quillen model categories?
More precisely, does there exist a pair (C,W) consisting of a category C and a subset W in Mor C (weak equivalences) such that (C,W) can be equipped with the structure of a Brown fibration (or cofibration) category, but not with the structure of a Quillen model category?
I would particularly be interested in examples in which the lifting axioms of Quillen are the obstacle. If they fail to be a Quillen model category just because they lack limits or colimits, I would be less enthusiastic.
On the other hand, I would also welcome examples in which it is relatively easy to show that they are Brown, but relatively hard that they are Quillen. 
I would also welcome good general stability properties. For instance, in the Brown case there are no big problems if you want to enlarge the set of weak equivalences, as long as the larger set satisfies (2 of 3) and as long as the resulting larger set of acyclic cofibrations is stable under pushouts (incision).
To summarise, I want to be able to exclaim: "Good that we have the Brown apparatus!"
 A: Otogonbayar Uuye observed in 1011.2926 that the category of $C^*$-algebras can be made into a category with fibrant objects in the sense of K.S. Brown in several ways. But there don't seem to be 
corresponding model structures, unless we extend the category of $C^*$-algebras.
A: First, let me add to Dai Tamaki's answer.
Indeed, the category of $C^\ast$-algebras admits several natural structures of category of fibrant objects. The easiest one takes weak equivalences to be homotopy equivalences and this describes the homotopy theory of $C^\ast$-algebras. And this does not extend to a model category structure as observed by J. Grodal (He actually considered a Baues fibration category structure). The reason he talks about suspensions and products is the following:
The category of commutative $C^\ast$-algebras is equivalent to the opposite category of pointed compact Hausdorff spaces. So there is a close parallel between topological spaces and $C^\ast$-algebras. By tradition, $C^\ast$-algebraists use the language of spaces for the analogues objects in the category of $C^\ast$-algebras. So tensoring by $C_0(I, \partial I)$ is called the suspension by $C^\ast$-algebraists, whereas homotopy theoretically it is really the loop-space (and loop-spaces should preserve products).
For an example "in which it is relatively easy to show that they are Brown, but relatively hard that they are Quillen", see nLab: Category of Fibrant Objects: Simplicial Sheaves:

Simplicial sheaves
The point of the axioms of a category
  of fibrant objects is that when
  passing from infinity-groupoids to
  infinity-stacks, i.e. to sheaves with
  values in infinity-groupoids, the
  obvious naïve way to lift the model
  structure from ∞-groupoids to sheaves
  of ∞-groupoids fails, as the required
  lifting axioms will be satisfied only
  locally (e.g. stalkwise).
One can get around this by employing a
  more sophisticated model category
  structure as described at model
  structure on simplicial presheaves,
  but often it is useful to use a more
  lightweight solution and consider
  sheaves with values in ∞-groupoids
  just as a category of fibrant objects,
  thereby effectively dispensing with
  the troublesome lifting property (as
  all mention of cofibrations is
  dropped).

In fact, one could say Brown developed his theory mainly to study this example.
A: In a series of papers starting with
Abstract homotopy theory in procategories, Cahiers Top. Géom. Diff., 17, 1976, pp. 113-124,
and with subsequent parts also in the same journal but published over the following two years, I explored a Brown type structure on a pro-category.  This is VERY much easier to define than the Quillen type theory.  There was both a fibration and a cofibration structure and they interact but do not seem to have a Quillen MC structure immediately there.  I think it is possible to generate up a Quillen structure (QMC) from either, but it is possible to develop quite a bit of theory without it just using the weaker structure.
The general question of Abstract Homotopy Theory is raised in the book:
Abstract Homotopy and Simple Homotopy Theory, World Scientific, 462pp (ISBN 981-02-1602-5) June 1997,
by myself and Heiner Kamps. We felt that there were homotopy theoretic situations in which the most
natural construction was via a cylinder functor or its dual cocylinder. The extent that a viable homotopy theory can be constructed then depends on filler conditions on that cylinder/cocylinder. We found conditions that gave a Brown type structure.
The theory as developed by Baues is similar but with a different aspect. Perhaps one should view the QMC approach as homotopical algebra whilst this approach was given the name Algebraic Homotopy and the two theories have slightly differing aims although often studying the same objects.
My own view is that 'the proper setting for abstract homotopy theory' is the one that works well in a given setting, as most of the time the importance of the theory is that it gives you theorems, calculations etc, in some application area, not results just for their own internal and intrinsic reason. Also what is `proper' at one time and for one development may not be for another time and another aim. There is no universal unique axiom system to be preferred to all others.  The QMC axioms are beautiful and the modern variants also, but the elegance of the Baues theory, the simplicity of cylinder based theories, etc. all have good points going for them.
A: Consider the category $Cat(S)$ of internal categories in a finitely complete category $S$ equipped with a Grothendieck pretopology $J$. For $S$ and $J$ satisfying certain properties, then there is a Quillen model structure, shown by Everaert, Kieboom and van der Linden where the weak equivalences are those internally fully faithful, essentially 'surjective' (*) functors. But without the asumptions that EKvdL give, there is only a Brown model structure in general (I don't know if this is published, Urs Schreiber mentioned it to me and I wrote down the proof myself). Mostly the problem comes with not having cocompleteness of $Cat(S)$, for which they give sufficient assumptions, like $S$ being a topos with NNO, or being a finitely cocomplete regular Mal'cev category. 
One might claim (Urs might, for example :) that this is a special case of a category of simplicial sheaves from Otgonbayar Uuye's answer, namely those simplicial sheaves that are nerves of categories and representable. But one does not need the machinery of simplicial sheaves to talk about this case.
A: Michael Weiss considers in his paper "Hammock localization in Waldhausen categories" the Waldhausen category of $G$-CW-spectra ($G$ a discrete group) where cofibrations are $G$-CW-inclusions and weak equivalences are simple homotopy equivalences, i.e. equivariant homotopy equivalences with trivial Whitehead torsion in $Wh(G)=K_1(G)/\{\pm g \mathrel{;} g\in G\}$. This class of weak equivalences is not strongly saturated, therefore it does not come from a model category. It would be very interesting to check whether this is a Brown category (probably easy) and whether some lifting axiom fails.
A: If you don't want to be forced to enlarge you may be interested in tha category of topological spaces and proper maps. They form a cofibration category (in the sense of any of the available definitions, I'd say) but there are not enough fibrations, i.e. no factorizations into trivial cofibration + fibration, etc.
A: The category of $L_\infty$-algebras with $L_\infty$-morphisms is a fibrant category, but does not have a Quillen model structure. 
Nevertheless this category is identified with the full subcategory of fibrant objects in a Quillen model category (the category $dgcu$ of differential graded counital cocommutative coalgebras). See this MO question. 
A: Consider an abelian category $A$ (or, more generally, an exact category in the sense of Quillen), then the category of complexes of $A$ is a category of cofibrant objects with the quasi-isomorphisms as weak equivalences and the degreewise (admissible) monomorphisms as cofibrations. This is not a Quillen model category in general (if you restrict your attention to bounded complexes, Quillen lifting axioms correspond exactly to the fact that there are enough injectives in $A$, which might fail in general, as can be seen by contemplating the opposite category of the category of sheaves over a sufficiently general topological space).
Also, even in the case when we have enough injectives, it is also possible to consider degreewise split monomorphisms as cofibrations (keeping the same weak equivalences), and then, Quillen axioms fail unless quasi-isomorphisms are all chain homotopy equivalences (in the case of an abelian category, this means that $A$ is semi-simple).
These examples are instances of a more general situation: consider a category of cofibrant objects $C$ with class of weak equivalences $W$. Then, for any class $S$ of maps of $C$, one can define a new class of maps $W(S)$ as the smallest one which contains $W\cup S$ and which satisfies the following properties: it has the two out of three property, and the class of cofibrations which are in $W(S)$ is closed under pushouts and finite sums. The good news are that $C$ is still a category of cofibrant objects with the same cofibrations but with $W(S)$ as class of weak equivalences.
This process is exactly what you need to define the notion of quasi-isomorphism of complexes of an exact category: starting from the category of bounded complexes with degreewise split monomorphisms and chain homotopy equivalences as weak equivalences (which is then a Quillen model category modulo the existence of finite (co) limits), one gets quasi-isomorphisms as the class $W(S)$ where $S$ consists of maps $X\to 0$, where $X$ runs over the family of complexes associated to admissible short exact sequences
$$0\to A\to B\to C\to 0$$
There is a non-abelian version of this construction: consider a (small) category $C$. We may then consider the category $s(C)$ of simplicial objects in the free completion of $C$ by finite sums. Then, considering the termwise split monomorphisms, there is a smallest class of maps $W$ such that $s(C)$ is a category of cofibrant objects with $W$ as weak equivalences and such that any simplicial homotopy equivalence is in $W$. The $(\infty,1)$-category (obtained by considering the Dwyer-Kan localization of $C$ by $W$) corresponds to the free completion of $C$ by finite homotopy colimits; for instance, for $C$ the terminal category, $s(C)$ is simply the homotopy theory of finite simplicial sets. If you do the same construction by replacing $C$ by its completion under small sums and
by replacing $W(S)$ by its closure under small sums and realizations, then you obtain the homotopy theory of cofibrant simplicial presheaves over $C$ (for the projective model structure), except that you didn't use any complicated tool to define it (no small object argument, no lifting theorem, in fact, you don't need to know the model category of simplicial sets at all). Of course, this apparent simplification comes at a price: you don't know how to construct homotopy limits in this language. The advantage is that you already are able to speak of homotopy colimits, so that you may use this to understand how to construct homotopy theoretic structures (e.g. Quillen model categories).
Another nice example of a category of cofibrant objects which is not a model category is the category of finite CW-complexes. One may argue that this is a subcategory of the category of cofibrant objects of a Quillen model category. But this is in fact always the case: any category of (co)-fibrant objects can be embedded very nicely in a proper simplicial model category; see Theorems 3.2, 3.10 and 3.25 and Remark 3.13 of my paper
Invariance de la K-théorie par équivalences dérivées, J. K-theory 6 (2010).
