Can a single set of conductors feed utilization equipment *and* a subpanel simultaneously under the NEC?
I'd like to see a real world example of this, but I don't think one could/would exist. 

If you had a piece of equipment that required a 30 ampere circuit, for example.  If you tapped a panel off that circuit, then you'd have to increase the size of the circuit, and/or follow the tap rules. The tap rules require overcurrent protection where the tap conductors terminate, so you'd need overcurrent protection at the new panel and the equipment.  

So basically, the branch circuit conductors would become a feeder, and tap conductors would be used to tap the feeder.

So no, I'm going to say you cannot tap branch circuit conductors. Maybe if it was a 15 or 20 ampere branch circuit, but what would be the point of that?