Now that I could lean back I guessed the answer for this puzzle after reading it a couple times mindfully. I think the answer is August 17. WHY? I will explain in detail. But first of all the answer I came across here (in NY Times) is obviously misleading and incorrect. They give July 16 as the answer, but their method is wrong because they eliminated both May and June and there isn’t enough information for such a step. What they should do is eliminate May 19 and June 18. Because these two dates (18, 19) are not duplicated in the list of dates.
Once they did so it would be clear from the conversation between Albert and Bernard that June 17 could also be eliminated. Because since Albert knows that June 18 and May 19 are not good candidates for DOB, neither can June 17 be. If June 17 was the date he would not claim “I don’t know the date of birth” because he already knows the month and there is only one date in this month that is June 17.