lolnole :
If the excess dust created thermal issues, then it will absolutely cause the symptoms you are describing. This is more apparent due to the form factor of the system (mini). Clean your computer to keep it cool!
Well crap, I turned to off to put the case cover back on and tighten the screws. Then it took it three tries to start itself.
How do I diagnose what's going wrong here?
There is plenty of airspace in and around the case. It's not crammed full of things, and the wiring is all neatly organized and tied to the corners of the rails.