1. August 17: Those who (a) assume Albert made Statement 1 on the assumption that Bernard didn't receive a unique date and thus didn't speak up, or (b) simply eliminated May 19 & June 18 by reasoning forward (as opposed to backward) from Statement 1 alone[1] (a rather common mistake)
2. July 16: Those who don't make the assumption in 1(a) or the mistake in 1(b)
July 16 is the better answer because it doesn't involve making the assumption.
Two crucial skills to get to the answer are (1) keeping perspectives separate among Albert, Bernard & a 3rd-party observer, and (2) enumerating each possible month (i.e. evaluating each premise) and playing out the statements in each case. People who don't take these actions end up saying, "There are 3 dates left at the end - how can you be sure it's July 16?!" They question how Albert can possibly know the date, even though we have to assume he's correctly concluded it. This is a perspective error; one has to put oneself in Albert's shoes, assume a month, and reason from there to see which assumption matches the observed statements.
As a Singaporean, I wish our question-setters had better grammar in general; it's tripped up test-takers before. And it hurts students' language skills by reinforcing grammatical mistakes.
[1] They reason forwards that "Albert says he knows Bernard doesn't know the date" ==> "Bernard doesn't know the date" ==> "The date isn't May 19 or June 18".
Instead, they should be reasoning backwards: "Albert says he knows Bernard doesn't know the date" ==> "Albert must have concluded this because of the month he was told" ==> "Albert must have been told a month with a 0% chance of Bernard knowing, i.e. a month with no unique dates".
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By the way, this problem has floated around for a while. Here's a 2006 occurrence of it:
http://mathforum.org/library/drmath/view/68613.html
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Date: 01/21/2006 at 00:43:08
From: Eddie
Subject: P -> Q <> ~Q -> ~P
A teacher told the month of her birthday (let's call it M) to Peter,
and the day of her birthday (let's call it D) to John.
Then, the teacher showed a set of dates, and asked Peter
and John which one was her birthday:
4 Mar, 5 Mar, 8 Mar
4 Jun, 7 Jun
1 Sep, 5 Sep
1 Dec, 2 Dec, 8 Dec
Peter said, "If I don't know the answer, John doesn't know, too."
John said, "At first I really didn't know, but now I know."
Peter said, "I know, too."
What is the teacher's birthday?