Is the "wire" you're talking about, actually armored cable?
When you look at it is it a metal spiral, with insulated wires inside?
Sounds like the "ground" wire you are refering to is the ground inside the armor sheath. Or at the very least it is flex conduit with a ground. Even though the flex conduit is metal and conducts well, it needs a ground wire because the spiral configuration creates a high inductance (if i remember correctly). Anyway it doesn't cut it to just clamp the flex conduit to the box.