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SPOILER ALERT:
Here is the solution I just came up with.
1) Albert says "I don't know when Cheryl's birthday is, but I know that Bernard does not know too."
If the birthday had been May 19 or June 18, then Bernard would have known it immediately after being told the day, because those days only appear once. In order for Albert to know this about Bernard's knowledge, he must know that the month is not May or June. (Becuase if the month was either may or June, Bernard might know the birthday right away). Therefore the birthday is not in May or June.
2) Bernard says: "At first I didn’t know when Cheryl's birthday is, but I know now."
After Albert gives Bernard the information that it is either July or August, Bernard learns the exact date. This indicates that the day is not the 14th, because the 14th is present in both July and August. In order for Bernard's information to not remain ambiguous, it must be one of July 16, August 15, or August 17, because those three have a unique day value out of the remaining possibilities.
3) Albert says: "Then I also know when Cheryl's birthday is." This indicates that for our three remaining possibilities, the month is not ambiguous, and therefore it must be July. (Because August had two possibilities left, while July had one). Therefore the birthday is on July 16.
I found it fairly easy, but I am experienced with logic puzzles, including those where characters in the puzzle use their knowledge about the knowledge of others in the puzzle in order to solve it. If someone wasn't experienced with these kinds of logic puzzles it would probably seem very hard. (The first time I encountered one of these I think I agonized over it for many hours).
Instead of writing spoiler alert using rot13 is better.