Node's cache isn't automatically cleared on update (intermittent problem)
I updated the APC module to 7.X-1.0-beta4 from beta3 and it seems to have fixed the problem. At least the client has not reported any further issues for several days now and we have not been able to reproduce the issue in our somewhat limited testing.

I noticed nothing in the change lot between beta3 and beta4 that would suggest a fix to this issue. I'm not convinced that this is what fixed it, but I have no other clues.