Question practically rewritten as from some answers&comments became clear that I am not understanding the semantics and context.
The "procedure" for Bonferroni is clear to me, my problem is a wrong and/or partial understanding of its semantics and how to use its outputs, so would need help with that.
To make easier to spot what I understand wrongly or miss I make a minimal example of my (mis)understanding and consequente perplexity.
John tests H01 on monday, gets a P-value of .04 and using the traditional alpha = .05 rejects h01.
On tuesday he tests h02, gets P-value .04 and rejects h02.
Bill tests the same two hypothesis together, gets the same p-values as Bill (.04 and .04), applies Bonferroni, alpha/2 = .025, his P values (.04 0.04) are > .025 so he retains h01 and h02 (while John with exactly the same P-values rejected h01 and h02).
This is what seems strange to me.
Original Question with some editing
(about testing N times the same hypothesis, which, I learned from answers, is not a correct use of Bonferroni) I think I understand the definition of Bonferroni correction but I do not see written explictly how one should reject the null hypothesis, so that is my naive question: when I perform N experiments, when do I reject the null hypothesis?
When one (or more) P-values are > alpha/N
when all P-values > alpha/N
when more than N/2 P-values > alpha/N