I have the same problem with minimum 8 iMacs. These have been updated to Mavericks this year. The mobile accounts existed since Snow Leopard, which has been the "starting" OS on most of the machines.
One new iMac with Mavericks pre-installed does not have the problem, the phd-account which this machine is using exists as long as the others.
OS X Server is running 10.6.8.
Last night I followed the guidance of Robert removing the open directory server in Users and Groups, unbinding all "home"-directories in Workgroup Manager and deleting the content of any home-directory of every mobile user account. We do not sync, it is only for using different ID´s.
After adding the open directory server again the client told me that the server does not support or deliver ssl. I accepted and added the ODS.
We never had this problem until OS X 10.9.4 update on Monday last week.