Albert and Bernard each have been given correct data.
Albert was given the month but, as there are multiple possibilities within each month, he can not determine the date.
Bernard was given the day. Possible days are: 14, 15, 16, 17, 18, and 19. Except for 18 and 19 there are multiple occurrances of each day.
Albert announces that he knows that Bernard can not know the date. This tells Bernard that the day is not unique. Therefore May 19 and June 18 can be ruled out.
There is now only one month with a single possible day: June. Therefore June 17
I like your logic (I think), but I keep arriving at July 16.
At last! I would have pondered this silly question for hours ...