I've found that this issue is unfortunately difficult to reproduce.
Pre-conditions:
1. The iPhone must be OFF.
2. At some point in time when the iPhone is off, you must receive a text message.
The problem is that (for me) the issue doesn't always happen for every txt message. It seems to happen more with non-Apple txt messages than with iPhone iMessages.
I have tested this several times but have not been able to reproduce on my own. I do this by turning my iPhone off, asking someone to txt me and then turning the iPhone back on, and in each of my tests, the messages show on the lock screen and then appear (as expected) in the Messages app. I have not identified the pattern for what causes certain txt messages to disappear (not show up in the Messages app). These tests have all been using iMessages sent from a different iPhone though.