I came up with the answer yesterday.
Here is my Explanation:
Answer is July 16
- 18 & 19 is out coz then Bernard would have known the bday as he knows the date and 18 & 19 occurs only once.
- May & June is out coz if its may or june ots possible for Bernard to know bday is 18 or 19. So the month has to be July or August
- Now Bernard knows the bday, means that it cannot be 14, so 14 is out.
- Now only 3 date is remaining. July 16, aug 15 and aug 17
- As there are 2 Aug, it cannot be august coz Albert would not have known bday. He knows means its July.
- As there is only one July its July 16

Enjoy !