Okay, I always get the phone models mixed here.

This is a black eight-button, sounds like a series E, but does it say "Dterm" at the top, or does it just say "NEC?"

What's probably happening is that one of the users is accidentally typing a feature access code for DND.

Another possibility is that you may have a key set as a DND button. Pressing the button, even by accident, will make it glow red, meaning that DND is set.

What kind of phone system is this connected to? IS it an Elite, an IVS/IVS2/IPS, IPX, etc. ... Somewhere in the building there's a big box on the wall (or if it's an IPX on the floor) and if you can describe it, including color, we might be able to tell you how to make this problem go away for good.