Michael
It helps to know that you are seeing a checkerboard - that is shown deliberately when an image cannot be accessed so you know ArcGIS attempted to read the file.
There's an older blog here Does your mosaic dataset look like a checkerboard? | ArcGIS Blog but it may not answer your problem.
In my experience, the checkerboard is caused by one of these situations:
1) user has run DEFINE OVERVIEWS but not BUILD OVERVIEWS (perhaps either forgot, or did not realize that "define" does not build). --> source imagery visible, but OVRs show checkerboard - clearly not your situation or you would never see overviews.
2) images used to exist but either got deleted or moved. Again not your situation or you would never see overviews.
3) user does not have permission to read from image directory --> this often exhibits the *reverse* of what you're seeing - e.g. overviews DO display (stored in a shared network location) but then when you zoom to full res, the original source imagery is locked away in a directory your IT manager has not granted access, so you see checkerboard @ full res. This also does not describe what you are seeing.
You indicate you can *always* access full res imagery, but sometimes you do see overviews, sometimes you do not. I don't know of any reason ArcGIS would care what else was loaded into the map, so the order of loading MDs should not introduce this problem. Are you confident that you never see a checkerboard when only one mosaic is loaded? Similarly are you certain there are never cases where you see a checkerboard, but then later that image succeeds?
It sounds like you have intermittent access to the storage location for the overviews. Can you verify where source datasets and also overviews are stored, and determine if there are any differences in read permission, max number of connected users, etc.?
