I killed my bear on May 31 this year and I believe that it had been over 20 years since I had killed one myself in May (and I kill at least one a year).
There is varying rutting behavior for most of June and that is a lot of fun.
In Quebec the season ends June 30. Most provinces bear seasons end anywhere from June 15th to June 30th.
Manitoba Stickflinger is correct when he says that most mature boars have nice fur at that time of the year. It is a phenomenon
that I noticed many years ago and that I mentioned to the biologists here in Quebec. As far as I know, it was not documented.
If you are thinking of going with Manitoba Stickflinger, in my opinion, Manitoba is as good as any place in North America to hunt black bears.