It's unclear to me both what counts as "abstract nonsense" and what counts as "looks like it should be provable by abstract nonsense", so I think the question is rather subjective. Nonetheless, I think it has merit, so here's a possible example under my own subjective interpretation. Taking inspiration from Aknazar Kazhymurat's comment, my own take is that one's understanding of "what is likely to be provable by abstract nonsense" should be constantly evolving based on new evidence, so maybe the question is really more like "what are some interesting examples of subtleties in the scope of abstract nonsense?".

Consider the category $Top_\Delta$ of Delta-generated spaces. This is the smallest full subcategory of the category $Top$ of topological spaces which is closed under colimits and contains the unit interval. Since it is generated under colimits by a small amount of data, it is natural to think that $Top_\Delta$ should be a locally presentable category (which is indeed the case). Indeed, until recently it was beleived that any cocomplete category with a small colimit-generator was locally presentable under the set-theoretical hypothesis known as Vopenka's Principle. So it was thought that, modulo set-theoretical subtleties, the local presentability of $Top_\Delta$ should be viewed as following directly from its definition.

Unfortunately, there turned out to be an error in the proof of this consequence of Vopenka's principle (and it turns out that $Set^{op}$ is a counterexample), so this line of reasoning is actually not valid. Even before this was realized, though, it was desirable to have a proof that worked without strong set-theoretical assumptions.

Perhaps surprisingly it turns out that the proof in the literature that $Top_\Delta$ is locally presentable relies on notions of infinitary logic. Now, in some sense, this just means that the proof is "doubly-abstract nonsense", but in another sense, it's a type of abstract nonsense that even sophisticated category theorists are not generally familiar with. An unwinding of this proof avoids explicitly using infinitary logic, but still requires one to think about topological spaces in terms of ultrafilters rather than open sets.

What is the moral of the story? I'm not quite sure. Maybe it's simply "accessibility questions sometimes require you to think hard and creatively about the specifics of your situation", even though they are often treated as an afterthought (I've certainly been guilty of this attitude before!).