I had broken the top screen on my previous 3DS, so I bought a new one and did a system transfer. I have a downloaded version of AC:NL on my SD card. I transferred the SD card from the previous system to the new one. Obivously, the new system started out at January 1st. When i changed the date in my 3DS to 9/11/2013, the game decided it was still Jan. 1st. So I changed the date in my DS to a random day, and checked back with Animal Crossing. I changed the date the 7/11/2013. This changed the date of AC:NL to 10/1/2013. Every time I changed the date, this would happen. If I changed it to 6/12/2013, the date in-game would be 9/2/2013.
I tried fixing it by backing up my data, deleting the game, then redownloading and restoring the game save, but this didn't work.
I'm out of ideas.