The bathtub waste overflow will not prevent a tub from overflowing. The tub overflow helps delay a bathtub from overflowing.
If a bathwaste is installed correctly along with a properly constructed Drain-Waste-Venting system, the overflow capacity can be as high as 14 gallons per minute. If the incoming flow rate of the tub spout exceeds the overflow's evacuation rate, then eventually water will go over the side of the tub.
In addition to delaying the tub from overflowing, the overflow is designed for snaking the tub drain. The evacuation rate of the overflow can be reduced by cavitation, which is air bubbles created by the tub spout stream pouring into the water close to the overflow plate. These air bubbles will prevent water from going through the overflow hole.
Even with a proper installation, the overflow feature will not prevent a tub from overflowing. To prevent water from overflowing the tub, never leave a filling tub unattended and turn off the water supply before the water level reaches the top.