This is an annoying bug I have found in the latest Audible for iPhone app. Audiobooks seem to download entirely but the fact they have finished doesn't seem to register with the system. The result is the app crashes when you try to play them (or on start up) and you can't delete the files from your library. Everything is fine however if you are offline.
This problem can be fixed by turning your phone into airplane mode in Settings, and then reopening the app. You can now play the audio files without the app crashing - but it is still not possible to delete them from your Library, it is however still possibly to delete them via the Now Playing / Last Played screen. Play the troublesome file and then click the button in the top right of the screen to bring up the Bookmarks/Details/Chapters screen.
Select the Details tab and scroll to the bottom of the page.
You can now remove the problematic audiobook from the device. You can then download it again after turning off airplane mode and everything should be fine.
This problem seems to occur when downloading from patchy connections and/or when the app is downloading in the background.