Multi-engine aircraft have enough power to successfully takeoff if one engine fails after V1. But what if, on a runway equal to the specified takeoff distance, someone decided to roll with one engine out? Are the airplanes specified be able to lift off before running out of runway?
You will not achieve book performance if you don't fly with book configuration.
See the portion of DeltaLima's answer here, including emphasis added by me:
[Takeoff Distance Required] has the aircraft accelerating to V1, suffering an engine failure at V1 and then continuing the take-off. This take-off involves accelerating from V1 to Vr, lifting off, retracting gear while accelerating and climbing, reaching V2 and 35 feet at the end of the runway. The runway length required for this scenario is the Take-Off Distance Required (TODR). The higher V1 the lower the TODR. This because when V1 is higher the part of the scenario that involves accelerating on a single engine becomes smaller.
These calculations take into account aircraft weight, engine thrust, aircraft configuration and runway condition.
Takeoff Distance Required is defined as the runway distance necessary to accelerate on all engines to V1, then experience an engine failure, then continue the takeoff to 35 feet above the ground.
It is self-evident that accelerating on less than all engines will require more runway than accelerating on all engines. Therefore you will not achieve the TODR numbers by starting your takeoff roll on one engine.