Matt,
This seems to be expected behavior. I downloaded your data but I am not seeing all of the labels show up, so I'm not sure if I am seeing what I'm supposed to be seeing.
At 100% in layout view, this is how it looks:
The way I saw it when I brought it in initially was like this (at 19% viewing):
When I look at the data view, initially the labels are not coming up (not sure if this is related to joined tables or not):
Turning on the Maplex Label Engine with "best" placement quality on labels, data view gave me this:
Maplex Label Engine with "best" in layout view (at 19%) shows this:
Zooming in on data view shows this:
Anyways, this, to me, is expected behavior. Quirk or not, this is normal.
A few things to take note of. Of the 58 counties in California, you apparently have over 23,000 features representing the same thing. This making the database huge and very slow. This is likely the reason why I didn't see all the labels (this could also be a factor of sharing a map package, I'm not sure without see the original data). I would suggest redoing this feature class and have it only contain 58 features, unless for some reason, you need more than that.
I am not sure if this is helpful or not but that's what I could determine.