TITLE: Non-examples of model structures, that fail for subtle/surprising reasons?
QUESTION [44 upvotes]: An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's limits lie, especially in cases where that's not immediately obvious.
Quillen model categories are a classic such case.  There are some easy rough intuitions—“something like topological spaces”, “somewhere one can talk about homotopy”, and so on—but various surprising examples show quite how crude those intuitions are, and persuade one that model categories cover a much wider range of situations than one might think at first.
However, I haven't seen any non-examples of model structures written up, or even discussed—that is, categories and classes of maps which one might think would be model structures, but which fail for subtle/surprising reasons.  Presumably this is because, given the amount of work it typically takes to construct an interesting model structure, no-one wants to write (or read) three-quarters of that work without the payoff of an actual example at the end. Has anyone encountered any interesting non-examples of this sort?

Background on my motivations: I'm currently working with Batanin/Leinster style weak higher categories, and have a problem which seems amenable to model-theoretic techniques, so I'm trying to see if I can transfer/adapt/generalise the model structures defined by Cisinski et al, Lafont/Métayer/Worytkiewicz, etc. in this area.  So I have some candidate (cofibrantly generated) classes of maps, and am trying to prove that they work; and there are lots of good examples around of how to prove that something is a model structure, but it would also be helpful to know what kinds of subtleties I should be looking out for that might make it fail to be.

REPLY [6 votes]: I've recently stumbled across another example, from 2017. It's in this paper by Wojciech Chacholski, Amnon Neeman, Wolfgang Pitsch, and Jerome Scherer. https://arxiv.org/abs/1702.05357v1
They are studying unbounded chain complexes over an abelian category $A$. For every injective class $I$ you can define a class of weak equivalences $f$ such that $A(f,W)$ is a quasi-isomorphism for all $W \in I$. You can then try to build resolutions, following Spaltenstein's work, via infinite towers where you truncate further and further to the left (i.e. more and more negative). If you have a model structure, you'll get these resolutions, but the authors get at them via "model approximations" instead, and give an example where even that fails.
In this paper, the authors formalize Spaltenstein's idea using a model category Tow($T$) of towers of successive truncations $\tau_{\leq -n}$. In Theorem 8.4 they give an example, using Nagata's "bad Noetherian ring," where the towers do not form a model approximation of $Ch(R)$. So, for this ring $R$, and for the choice of injective class in Section 8, $Ch(R)$ doesn't have a fibrant replacement functor built in the usual way. (and very likely, not one at all with respect to that injective class). The failure has to do with infinite Krull dimension, which I hope qualifies as "subtle and surprising."