so the first statement is the first clue. neither one knows. That means if there is only 1 choice for day, those are eliminated. because they do not know means there is more than 1 option.
that eliminates 18th and 19th days from the running, because they only occur in 1 instance.
if Bernard knows the day, which he does, if it were only in 1 instance, he would automatically know the month. he does not, so the day of the birthday is either 14, 15, 16, or 17.
Since eliminating 18th and 19th days only leaves 1 option for June, and Albert does not know the month, June is eliminated.
I'm totally stuck after that point.
I think the answer is June 17th.