To close the loop on this...
After much troubleshooting and gnashing of teeth we finally identified the source of the problem.
We are running ArcGIS Server 10.2 with a pooled clustered configuration consisting of 2 web adapters machines with a load balancer. The extent errors we were seeing were actually the result of one of the two web adapter boxes spitting out a different spatial reference than the one defined for the feature service.
In our case the service was configured in Web Mercator based on a geodatabase in WGS84. Ordinarily this works fine, but in our case at some unknown point in time after starting the service one of the web adapters would start reporting the extent of the service in Lat/Long instead the correct Web Mercator coordinates. This was apparent when looking at the REST endpoint; every refresh would alternate between WGS84 and Web Mercator as the load balancer dutifully alternated requests between the two machines.
As a result our users would basically have a 50/50 chance of pulling down a database in which they could save data. If they pulled a database from the server with the bad definition, they ended up with an offline geodatabase with a writable extent in the immediate proximity of NULL island.
What is causing the fail-over machine to get out of whack is still a mystery. This bug has cropped up in at least 3 projects we know of, and forced us to remove the fail-over machine from our production environment... severely compromising the system's capacity. The bug has only appeared in projects with the WGS84 database/Web Mercator Feature service configuration. Projects where the database is in State Plane don't seem to exhibit the bug.
This appears to be a fairly significant bug with ArcGIS Server, does anyone know if this issue will still exist when we migrate the platform to 10.3?
Edit: fixed terminology