I'd be careful answering this because apparently it's a big deal on social media and some really smart guys were debating it or whatever. But this is what I know.
There are only two dates with unique numbers: May 19 or June 18. If Bernard was given a date of either 18 or 19, he'd know right away. The fact that Albert says Bernard doesn't know means the month he was given wasn't May or June. Bernard has now figured this much out, and if he had been told 14 he wouldn't know the answer as there are still two 14s left (July and August) leaving July 16, August 15 and August 17. Bernard must've been given the number 16, because for Albert to say he knows too would rule out August, which still has two remaining dates. That leaves the answer: July 16.
