How can we possibly know all possible test when applying the Bonferroni correction? I never quite understood the rationale behind the Bonferroni correction. I understand that we want to reduce the FDR, but it seems odd to penalise a test based on the existence of another test, given that we do not even know if we have complete information about all possible test.
As an example, say we perform some test on time series data (let's say, the Mann Kendall test). Let's say we have temperature data about New York from 1900 to 2020. If we perform the test for the full dataset, from 1990 to 2000, that gives us one test. We find this is significant. We could now perform another test, but with a subset, using data from 1950 to 2020. We also find this is significant, but we now apply a Bonferroni correction. We now perform another test, from 1990 to 2020, and now the original test (1900-2020) is no longer significant from the correction. We could continue like this until we have no significant results, even though we haven't changed the raw data.

Bonferroni only says: If one has a set of $n$ independent hypothesis tests $\{T_i\}_{i=1}^n$ each with a belonging null hypothesis $H_{0i}$, and one wants the probability of the event, that at least one of those tests has a false discovery, to be less than $\alpha_a$, it is sufficient to make sure that the probability of false discovery of any of those tests is less than $\alpha_a/n$:
$$
\forall_{i=1,\ldots,n} P(T_i(x) = 1 \;|\; H_{0i}) \le \alpha_a/n \quad\Longrightarrow\quad P(\bigvee_{i=1,\ldots,n} T_i(x) = 1 \;| \bigwedge_{i=1,\ldots n} H_{0i}) \le \alpha_a.
$$
Thus, you do not penalise a test based on the existence of another test, but you rather consider a completely different event, the compound event of all the tests together to have no false discovery. This is like when you have two coins: If you want the probability of the first coin to come up tails to be $1/2$, then this will be accomplished by a coin that is balanced ($p_{tails} = p_{heads} = \frac{1}{2}$), and this will not be changed or penalised by what the second coin might come up with. But, of course, if you consider the compound event of both of the two coins to come up tails, and you want the probability of this compound event to be below $1/2$, too, then one way to accomplish this would be to require the coins both to be unbalanced with the probability for tails to be below $1/4$ for both.
Alternatives to Bonferroni: While Bonferroni is clearly correct (and the bounds obtained by dividing by $n$ are rather tight), the problem of the OP still remains: if we are given many tests, how should we combine them into a super-test that doesn't have the disadvantage of the above method that, if there are too many tests, there will never be any discovery, even though there actually would be something to discover? Fortunately, there are alternative procedures. One is the Benjamini–Hochberg (BH) method: In a nutshell, the approach is not to create a single compound hypothesis test. Rather, BH returns those tests $\{Q_i\}_{i=1}^m$, that are most likely to really represent discoveries. For that, BH has a parameter $q$ which we can choose, and then BH guarantees that the ratio of wrong discoveries in the returned test set $\{Q_i\}_{i=1}^m$ will be less than $q$.
