Are cofibrations accessible? The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations?
More generally, let $C$ be a locally presentable category and let $(L,R)$ be a weak factorization system on $C$. If $(L,R)$ is cofibrantly-generated (i.e. there is a set $I \subseteq L$ such that $R$ consists precisely of those morphisms with the right lifting property with respect to $I$), then $R$, considered as a full subcategory of $C^\to$, is accessible and accessibly embedded.
Question 1: Suppose that $(L,R)$ is cofibrantly-generated. Is $L$ accessible and accessibly embedded (as a full subcategory of $C^\to$)?
Question 2: Conversely, if $L$ is accessible and accessibly embedded, then is $(L,R)$ cofibrantly-generated?
Question 3: Similar to the above two, but use the notion of "small-generated" coming from Garner's small object argument (where $I$ can be a category rather than a set).
The proof that $R$ is accessible and accessibly embedded is not completely straightforward: it relies on the fact that the small object argument provides a functorial factorization system which preserves $\lambda$-filtered colimits and $\lambda$-presentable objects for some $\lambda$ to exhibit every $R$-morphism as a retract of a colimit of $\lambda$-presentable $R$-morphisms and to see that fibrations are closed under $\lambda$-filtered colimits.
The fact that $L$ is closed under transfinite composition sounds tantalizingly close to saying that it is closed under filtered colimits, but I'm not sure the latter is actually true.
Motivation:
If the answer to both questions is yes, then it becomes very easy to prove Jeff Smith's theorem since an intersection of accessible, accessibly-embedded, replete subcategories is accessible and accessibly-embedded.
 A: A cofibrantly generated $(L,R)$ does not need to have $L$ accessible, see Example 3.5
in my paper "On combinatorial model categories."
Also, $L$ accessible does not imply that $(L,R)$ is cofibrantly generated, even accessible.
Take regular monos in Boolean algebras. This $L$ is accessible but $(L,R)$ cannot be accessible because
regular injectives are complete Boolean algebras which are not accessible.
A: Here's an elaboration on the example in Professor Rosický's paper. I'll make it community-wiki.
Let $Pos$ be the category of posets, and let $L$ be the class of split monomorphisms in $Pos$. Let $L_\omega$ be the set of split monomorphisms between finite posets.
Claim 1: $L$ is the cofibrant closure of $L_\omega$.
Proof: One can check that in any category the class of split monomorphisms is closed under coproduct, cobase-change, transfinite composition, and retracts. Conversely, if $P \to Q$ is a split mono, one can add the elements of $Q$ one at a time in a chain, so we may assume without loss of generality that $Q$ has only one element $q$ which is not in $P$.  Now we may express $P \to Q$ as the colimit of a chain, each link of which adds one relation $p \leq q$ or $q \leq p$ for some $p \in P$. Each of these links is a pushout by a split mono between 2-element posets. I'm not sure how do do this!
Claim 2: $L$ is not closed in $Pos^{\to}$ under $\lambda$-filtered colimits for any $\lambda$.
*Proof:** The closure of $L$ under $\lambda$-filtered colimits consists of the $\lambda$-pure monomorphisms in $Pos$. So we just need an example of a $\lambda$-pure monomorphism which doesn't split, for each regular cardinal $\lambda$. The inclusion $\lambda \to \lambda+1$ fits the bill -- see Example 2.28(3) in Adamek and Rosicky's Locally Presentable and Accessible Categories.
Thus $L$ is cofibrantly generated, but not accessibly embedded.

In the other direction, I don't know a source for Professor Rosický's claim that regular monos in Boolean algebras are a counterexample. But I'm pretty sure that in any locally presentable category, both (epi, strong mono) and (strong epi, mono) are accessible orthogonal factorization systems. And Example 4.4(2) in the same book says that complete Boolean algebras are the injective objects in the category of distributive lattices, citing 
Banaschewski, B. and G. Bruns (1967): Categorical characterization of MacNellie completion. Arch. Math. 18, 369-377.
I think it's well-known that complete Boolean algebras don't form an accessible category. To show this it suffices to construct a Boolean algebra of cardinality $\kappa$ which is $\kappa$-complete but not $\kappa^+$ complete, for arbitrarily large $\kappa$. The set of $<\kappa$-sized subsets of a set of size $\kappa$ works (where $\kappa$ is regular).
