Brandon Simnacher has the correct answer on how to get around this, but I wanted to post why this is happening. I see this as one of two potential issues.
Started Playing Online, Continued Playing Offline
The issue is a by-product of saving games while connected on- or offline. When your system is offline and you save a game, it reverts to the release date for the hardware, in this case November 21, 2005. So what might have happened was you started playing Bioshock Infinite online, giving autosaves a date in April of 2013, and then continued playing offline, giving your autosaves a date of November 21, 2005. The Continue option looks for the most recent date, which in your case would have been the last time you were online. This would also be earlier in the storyline for you.
- You play chapters 1-8 while your Xbox is connected to the internet. All saves dates are in April 2013.
- You continue the next day while not connected to the internet, making it through chapter 12. Chapters 9-12 have save dates of November 21, 2005.
- You go to "Continue" your game the following day, but Bioshock sees that your most recent save date was in April 2013, taking you back to the beginning of Chapter 9.
The best solution so that this doesn't happen is to make sure that you are always online when playing the game (giving all of your saves sequential date information). If this isn't an option, navigate to the Load option instead of using Continue.
Always Playing Offline
This may be a problem of never being online as well. If you are never online, every time you turn on your system it will think it is the same day of 2005 (sometime in November IIRC). So if you play one day starting at noon and go for four hours, you'll get save times ending at 4pm on November 21, 2005. When you come back the following day, you start playing at noon and play for 2 hours, you'll get save times ending at 2pm on November 21, 2005. How your saves would look to Bioshock Infinite:
- Day 1: Play for four hours, last save is at 4pm on November 21, 2005.
- Day 2: Hit continue, game loads save from 4pm on November 21, 2005. Play for 2 hours, last save is at 2pm on November 21, 2005.
- Day 3: Hit continue, game loads save from 4pm on November 21, 2005 since that is the most recent save date.