--> Confirm that your call is hitting the right call handler. --> Check the greeting is uploaded correctly. --> Check only standard is enabled if any other will be enabled like alternate it will take priority over standard greeting.  

So as you mentioned that you have done some reconfiguration and it's hard to tell without seeing the logs that what is the exact issue it can be the commands or configuration were run incorrectly and in partial creating corruption and unwanted change...

Hi Mumbles,   As you mentioned the problem is occurring on specific time. This type of fault mostly occurs because of miss-configuration on the Call Handlers. You should check your call handler configurations.    

Try to Erase ITL and CTL files from phones.