Say your paper has four independent tests of your hypotheses. Suppose that in fact all your hypotheses are false. Then, under the null, your p-values are uniformly distributed between 0 and 1. What's the chance of getting at least one result at 5% significance?
We can answer this with a one-liner in R:
You'll get a significance star about 18% of the time.
What about if you have eight hypotheses?
> table(replicate(100000, all(runif(8)>0.05)))
About one third of the time.
Multiple hypotheses really affect your p values even if you just test a few hypotheses. This is not just a problem for people using genetic data and running millions of tests! But you almost never see a paper which corrects p values for multiple hypotheses. Perhaps this should change.