rd-aaron has it. I disagree with Polak's solution.
Spoiler!
Cheryl tells Albert the month she was born. She tells Bernard the day.
Neither Albert or Bernard can figure it out from the info she gave.
When Bernard hears Albert say that "I don't know, and I know Bernard doesn't either", then Bernard knows the answer.
From this statement we know that it cannot be June 18 or May 19. (Because, Bernard, knowing the day would have figured that out on the spot as both 18 and 19 are unique in the puzzle.)
At this point, Bernard already knows is not May or June (we don't know, but that, but we can reason it out from Albert's statement that "I don't know, and neither does Bernard.) Albert's statement gives us some extra info that we didn't know in the beginning. Now we ALL know its NOT May or June.
That leaves July and August - both of which have days in common, except for the 16th.
Bernard can then figure out that the only non-repeated day is the 16th, so her birthday must be July 16.
And when Albert hears Bernard say that he knows (based on Albert's previous statement), that gives Albert the missing piece of the puzzle too.
I think the question is poorly worded, grammatically speaking. Kind of a mean thing to do to kids in a high-pressure test situation. Its not really clear that Bernard and Albert are able to hear each other. But then again, maybe that's a cultural assumption in Singapore?