Maybe it's good to explain the ''reasoning behind'' multiple testing corrections like the one of Bonferroni. If that is clear then you will be able to judge yourself whether you should apply them or not.
In a hypothesis test one tries to find evidence for some known or assumed fact about the real world. It is similar to ''proof by contradiction'' in mathematics, i.e. if one wants to prove that e.g. a parameter $\mu$ is non-zero, then one will assume that the opposite is true, i.e. one assumes that $H_0: \mu=0$ and one tries to find something that is impossible under that assumption. In statistics things are rarely impossible, but they can be very improbable.
So if we want to show that $H_1: \mu \ne 0$ then we assume the opposite namely $H_0: \mu = 0$ and we try to find something very improbable. Very improbable is defined in terms of a probability lower than an a priori fixed significance level $\alpha$. Note that, because of the analogy I will use terms such as ''statistically proven'' or ''statistical evidence'', these terms aree just used for didactical reasons and are not used in general.
In order to find that ''low probability'' we draw a random sample form a distribution that is known when $H_0$ (our assumption of the ''opposite'' of what we want to prove) is true. As we assumted $H_0$ te be true we can compute the probability of this outcome (more precise something that is at least as extreme as this outcome).
As the sample is a random draw from a distribution, it may be that we obtain a low probability just by ''bad luck with the sample'' and then we reject $H_0$ just because we had bad luck with the sample. Rejecting $H_0$ means that we consider to have found evidence for $H_1$ but it is false evidence in these cases where we have bad luck with the sample.
False evidence is a bad thing in science because we believe to have gained true knowledge about the world, but in fact we may have had bad luck with the sample. This kinds of errors should consequently be controled. Therefore one should put an upper limit on the probability of this kind of evidence, or one should control the type I error. This is done by fixing an acceptable significance level in advance.
So if we fix our significance level at $5\%$ then we are saying that we are ready to reject $H_0$ when it is true (because of bad luck with the sample) with a chance of $5\%$. As (see supra) rejecting $H_0$ is ''statistical evidence'' for $H_1$ this means that we falsely consider $H_1$ as ''statistically proven''.
Assume now that we have two parameters, and we want to show that that at least one is different from zero. Follwing the logic of ''proof by contradiction'' we will assume $H_0: \mu_1=0 \& \mu_2=0$ versus $H_1: \mu1 \ne 0 | \mu_2 \ne 0$ and that we use a signficance level $\alpha=0.05$.
One possibility to do this is to split this hypothesis test and to test $H_0^{(1)}: \mu_1=0$ versus $H_0^{(1)}: \mu_1 \ne 0$ and to test $H_1^{(2)}: \mu_2=0$ versus $H_1^{(2)}: \mu_2 \ne 0$ both at the significance level $\alpha=0.05$.
To do both tests we draw one sample , so we use one and the same sample to do both of these tests. I may have bad luck with that one sample and erroneously reject $H_0^{(1)}$ but with that same sample I may also have bad luck with the sample for the second test and erroneously reject $H_0^{(1)}$
Therefore, the chance that at least one of the two is an erroneous rejection is 1 minus the probability that both are not rejected, i.e. $1-(1-0.05)^2=0.0975$, where it was assumed that both tests are independent. In other words, the type I error has ''inflated'' to 0.0975 which is almost double $\alpha$.
The important fact here is that the two tests are based on one and the sampe sample !
Note that we have assumed independence. If you can not assume independence then you can show, using the Bonferroni inequality$ that the type I error can inflate up to 0.1.
Note that Bonferroni is conservative and that Holm's stepwise procedure holds under the same assumptions as for Bonferroni, but Holm's procedure has more power.
When the variables are discrete it's better to use test statistics based on the minimum p-value and if you are ready to abandon type I error control when doing a massive number of tests then False Discovery Rate procedures may be more powerful.
EDIT :
If e.g. (see the example in the answer by @Frank Harrell)
$H_0^{(1)}: \mu_1=0$ versus $H_1^{(1)}: \mu_1 \ne 0$ is the a test for the effect of a chemotherapy and
$H_0^{(2)}: \mu_1=0$ versus $H_1^{(2)}: \mu_2 \ne 0$ is the test for the effect on tumor shrinkage,
then, in order to control the type I error at 5% for the hypothesis $H_0^{(12)}: \mu_1=0 \& \mu_2 = 0$ versus $H_1^{(12)}: \mu_1 \ne 0 | \mu_2 \ne 0$ (i.e. the test that at least one of them has an effect) can be carried out by testing (on the same sample)
$H_0^{(1)}$ versus $H_1^{(1)}$ at the 2.5% level and also $H_0^{(2)}$ versus $H_1^{(2)}$ at the 2.5% level.
