I've observed something interesting lately, relating to apps keeping information when in the background, with an app called gMusic. This plays Google Music content, either streaming or offline.
If you 'minimise' the app, then for a short time it will re-open at exactly the place you left it (as the native music app does, and everything else pretty much). If you re-open it after a while, it opens on the 'artist' list, but when you click on the 'now playing' button it returns you to the exact second of the exact song you were listening to. This is regardless of how long you've left it 'minimised', and what you've done since.
It's not the same as it used to be on iOS 5, where it would simply open where you left it, but it indicates that it is possible to keep this information, which even the native app loses (you start at the beginning of the song you were listening to).
Even more interestingly (maybe), it keeps your place even when the app is completely closed, and for however long between listens. I've killed it (on iOS 7) and opened again weeks later, and the 'now playing' tab restores the precise place I'd been listening.
So it's possible.