Yes.
Both terminals of the regulator DO NOT go to the alternator.
Yes, they do
the ign + / field wire turns the regulator off / on and makes power available for the field windings. It also serves as a "sense" line to tell the regulator how much current the battery / system needs to maintain the existing load
The other field connection applies a variable ground to the field windings which in turn controls the alternators out put.
So, based on the above you either have
1. A bad regulator
2. a bad ground at the regulator
3. A pinched field wire between the regulator and alternator
4. A bad alternator/ ground.
NOTE: if 1-3 do not apply, I suspect one of the field terminals is grounded to the case