[Oling_Roachinen]

Spoiler!

Hmm, because Albert knows Bernard doesn't know, we know that the month is July or August, as May and June have unique days. Bernard uses that and is able to solve, so it can't be the 14th of either month. That leaves July 16, August 15 and August 17. Since Bernard figures it out when Albert knows it, we know it can't be August because it has 2 answers. So July 16?

Edit: Took a bit longer than rd_aaron, but I got the same :P