To solve the puzzle, we can break down the clues given:

1. Cheryl gives a list of possible dates:
- May 15, May 16, May 19
- June 17, June 18
- July 14, July 16
- August 14, August 15, August 17

2. Albert knows the month, and he states that he knows Bernard doesn’t know either. This implies Cheryl's birthday cannot be in May or June, because if it were, there are unique days (May 19 and June 18) that could lead Bernard to potentially know.

This leaves us with:
- July 14, July 16
- August 14, August 15, August 17

3. Bernard then says at first he doesn't know either, but now he knows. This means the day must be unique within the remaining options:
- July 14 (could also be August 14)
- July 16 (unique)
- August 15 (unique)
- August 17 (unique)

This excludes July 14 and August 14 since they are not unique.

Now we have:
- July 16
- August 15
- August 17

4. Finally, Albert hears Bernard and he states he now knows when Cheryl's birthday is. This implies that the month he has must correspond to a unique day left:

If the month were August, it would be unclear whether it is August 15 or August 17. Therefore, the birthday must be July 16 because it is the only date left that suggests a unique day for Albert after Bernard's statement.

**Cheryl's birthday is July 16.**