That's a neat question !
I found a Knowledgebase document that explicitly says that's not possible with the 1756-L7x controllers, and I don't expect it to be possible with any other controller platform.
I've done it with a 1756-DNB... but it was a feature specifically of that module and was in the tatus controller-scope tags.
Ordinarily the only thing scrolling on the display is the program name. Most other faults can be separately monitored with GSV or MSG instructions (major and minor fault codes).
If there was a controller OS crash that showed one of the self-diagnostic messages, of course the user program wouldn't be running so programmatic access would be moot.