When (in the context of 'non-independent tests') does Bonferroni correction *not* yield the appropriate FWER? I have a question about the oft-heard suggestion that a Bonferroni correction is appropriate in the context of posing multiple independent hypotheses and inappropriate elsewhere, in contrast to statements like:

One of the most widespread of these misunderstandings is that the method would be based on an assumption of independence between p-values. This misunderstanding comes from a frequently used, but deficient motivation for Bonferroni, saying that the probability of making a false rejection if all $m_0$ p-values of true hypotheses are independent, and we perform each test at level... By this reasoning, Bonferroni seems like a method that only provides approximate FWER control and that requires an assumption of independence for its validity. In fact, the method of Bonferroni provides exact FWER control under any dependence structure of the p-values.

To explore this question numerically, I wrote a small R script where I simulated multivariate normal data according to some correlation matrix (sampled from a flat LKJ), estimated sample correlations among that data, and computed the corresponding p-values. I then estimated regression coefficients between these samples $x$ and independently generated data (i.e. from $y = a + 0 \cdot x + e$), and then computed the corresponding p-values. After Bonferroni correction, the FWER for both was at the desired value, $\alpha$. This should also apply if the correlation structure / non-independence were in $y$, given the symmetry between $y \sim x$ and $x \ sim y$.
So is this whole 'non-independence between tests' thing actually a thing w/ the Bonferroni? I'm aware there are other motivations for disfavoring the correction, e.g. it only bounding the FWER from above, and with sufficiently many tests over finite data you get ever higher false-negative rates as the significance threshold goes to 0. If the dependence structure between tests is irrelevant to Bonferroni correction, where did this advised requirement of "multiple independent hypotheses" come from?
(sorry if this question is ill-posed -- maybe I'm misunderstanding the term 'independence' here. I usually just fit hierarchical models, cite Gelman, Hill, & Yajima '12, and call it a day, but am working more in a NHST world lately and am trying to wrap my head around different multiple comparisons strategies)

Assuming that the marginal tests all are valid (i.e., $P_{0i}(p_i \le \alpha) \le \alpha$, for all $\alpha \in (0,1)$), then the Bonferroni correction guarantees that the FWER is $\le \alpha$ as well, for all dependence structures.  In other words $P_0(\min p_i \le \alpha/m_0) \le \alpha$.  Here, $P_0$ refers to the complete null model, but the result also applies for partial null models, when the minimum is taking over the p-values for hypotheses that are in the subset of true nulls.
Since there is a "$\le \alpha$" in the result, it hardly sounds like one could call this result "exact."  "Valid" would be a better term.
On the other hand, you can get exact results, ones that replace "$\le \alpha$" with "$= \alpha$," by exactly exploiting correlation structures, e.g., as in Tukey's method. Such methods reduce the inherent conservatism of the Bonferroni method. You can also make the results approximately (asymptotically) $= \alpha$, rather than $\le \alpha$, again reducing the conservatism, by approximately exploiting correlation structures (e.g., using the bootstrap).
Here is a reference.
