It would be worthwhile to run callviewer and port status monitor when you call into Unity. Both are available via the Unity Tools Depot icon on the desktop of the Unity server.
Be sure to hit the double play button on port status monitor to monitor all voicemail ports. Use the call viewer program to see a realtime summary of all calls hitting the system, look for your calling number and reference the port ID that the call is coming in on. Having the port id handy is useful on a busy system so you can look at the right port on port status monitor.
You should be able to log or copy/paste the output from that to here, but like pndscm said it sounds like a scheduling problem. The Opening Greeting call handler generally should not have any schedule other than the 'all hours, all days' schedule, I would think.