Yes we do. Marina stole our contest with an early entry. She knew that Casablanca took place in December 1941, right before Pearl Harbor.
How can the viewer tell? It's in the famous scene where Rick (Humphrey Bogart) is getting drunk after Ilsa Lund (Ingrid Bergman) has come into his bar with Victor Laszlo (Paul Henreid). This is the scene where Bogart says, "Of all the gin joints in all the towns in all the world, she has to come into mine." (He does NOT say, "Of all the joints in all the towns..." it's "gin joints." And he never says it to Ilsa.) But I digress. As Bogart is drinking and feeling sorry for himself, Sam (Dooley Wilson) pulls his piano over and starts to noodle. (Wilson did not actually play the piano.) Bogart says to him, "Sam, if it's December 1941 in Casablanca, what time is it in New York?" Now, we know that the US is not in the war--the whole point of the film is that Rick/Bogart stops being neutral and gets into the fight on the anti-Nazi side., and we know that Pearl Harbor was bombed on December 7th. Hence, the film takes place during the preceding week.
Actually, if you are VERY, VERY sharp-eyed, you can tell that the story occurs on December 2nd, 3d and 4th. We first see Rick in an early scene, after the croupier brings him a marker to approve; the marker is put down on the table, the camera follows it, a hand with a pen comes into the shot and signs the marker, then drops the pen and moves to a cigarette. The camera then follows the hand holding the cigarette up and we see Bogart's face. At the beginning of this sequence, if you look very carefully, you can see that the marker is dated "2 Decembre 1941," December 2nd.
One of the reasons why Casablanca is such a great film is shown in the first scene discussed above. After Rick says, "...if it's December 1941 in Casablanca, what time is it in New York?" he answers his own question: "I bet they're asleep in New York. I bet they're asleep all over America." The first part is literal--it's 3 or 4 in the morning, maybe later, in Casablanca, so people are in bed in the Eastern United States. But when he says "I bet they're asleep all over America," that's metaphor.
Marina, by the way, also answered our extra-credit questions correctly, which makes me suspect that she has a copy of Richard J. Anoblie's Casablanca, a collection of stills from virtually every shot along with a running script. (The script was done from transcription, not from a printed script. If you want to know how I can tell, email firstname.lastname@example.org.)
And, Marina, if you'll email the Old New Englander, you can find out how to collect your prize, a lovely bumper sticker.
Thanks to all who entered.