iOS Devices Stopped Syncing

We have a large subset of mobile devices, mostly iOS, that just stopped syncing e-mail with their Exchange accounts. 

We have thoroughly searched the logs across all servers in the 2010 and 2013 environments and have not been able to find the root cause of this problem within the e-mail system.  With nothing more to go on, we performed a full restart of all 2010 CAS and all 2013 servers, which does not appear to have corrected the issues.

We do see requests to //server/Microsoft-Server-Activesync in the IIS logs for some of the affected users, with a status of 200, but the “LastAttemptTime” values for these devices remains several days behind.

All connectivity tests (testconnectivity.microsoft.com, test-activesync cmdlet, and browser checks)  come back looking good.  The majority of devices in our ecosystem can sync fine, but a fairly good sized subset cannot.  Some of the affected devices can be fixed with a restart, while others require a full removal of the Exchange account.

I have attempted to turn on verbose ActiveSync logging for some of the affected accounts, but since they do not sync, I am not able to get anything.

If anyone has any suggestions as to what might be going on here, it would be a huge help.