If you rob some new potatoes carefully without disturbing the plants too much, they will continue to mature the ones already started.

I would say that there is a limit to how high to hill. The potatoes are going to call a halt sooner or later to continued growth and they will start to die down. Also more tubers means smaller ones.

The potato plant sets new potatoes on stolens. The stolens grow from the seed potato to the first new potato to the next ...

O--o--o--o

If you pick the first new potato, you're picking all the potatoes that would have formed from that stolen.

It won't prevent the plant from growing new stolens.