If an object is tied down or experiencing a restraining force, the net force acting on the object will be the difference between the applied force and the restraining force. If the restraining force is greater than the applied force, the object may not accelerate or may even move in the opposite direction. Therefore, the statement "The greater the force that is applied to an object of a given mass, the more the object will accelerate" would still hold true, but the acceleration of the object may be limited by the restraining force.
