Digital volt meters pickup stray voltage and can cloud diagnosis. Always use a incandescent test light not a LED as LED can pick up strays.

Basically there is a high resistance in the circuit that makes it look like you have voltage without a load but when a load is placed on the circuit, there isn't enough amperage.

Think of a garden hose that has a kink, the pressure will be high when the nozzle is closed but as soon as you open it, pressure drops and flow is minimal.