We are using the new collector app, not the old one that didn't support labels.
So I have labels set up for something we call a "location ID" (a 1- or 2-digit number, showing when zoomed out a bit to "building" level) and also for poles (10-digit number, that appear when zoomed way in, to "room" level). Ideally when a person zooms into a location, he or she can see the location ID alongside the pole number.
However I have noticed - and this was not a problem a couple months ago, when I last uploaded layer data for Collector - that our location IDs disappear when zooming in. The pole numbers come in as normal, at the visibility level I set up, but the location IDs disappear well ahead of the pole labels coming on, so there is a zoom level or two where there are no labels at all.
After testing the only way I can get the location ID labels to remain visible is to turn off the pole number labels. Obviously the two labels are interacting badly, and hopefully someone can help out!
The workaround I have discovered is to add a new field with all NULL values. For example, I added a field to my poles called "noLabel" and set the labels to that. When I save the map and then download it, I no longer have pole numbers obscuring or affecting my service numbers.
As for the glitch, I think it's because I published the service layers from ArcMap with pole number labels turned on. I noticed that when I turn off the pole number labels in the AGOL map viewer, the pole number labels still show up on Collector's downloaded/offline map. When I changed the labels to another field, those other values show up as labels in Collector. But when I turn off labels again and save, the labels still show in the offline map. So I can change the labels that show, but I can't turn any of them off. Thus the creation of a new, NULL value field and having my poles be labeled on that.
Furthermore, the layer label settings reset. I may have turned labels off for the poles, but after I save, leave the map, and return to it, I see that the poles have labeling checked on again. So whatever the problem is, it seems that I can't override that "Label Features" check-box.