Teleoperation is increasingly recognized as a viable solution for deploying robots in hazardous environments. Controlling a robot to perform a complex or demanding task may overload operators resulting in poor performance. To design a robot controller to assist the human in executing such challenging tasks, a comprehensive understanding of the interplay between the robot's autonomous behavior and the operator's internal state is essential. In this paper, we investigate the relationships between robot autonomy and both the human user's cognitive load and trust levels, and the potential existence of three-way interactions in the robot-assisted execution of the task. Our user study (N=24) results indicate that while the autonomy level influences the teleoperator's perceived cognitive load and trust, there is no clear interaction between these factors. Instead, these elements appear to operate independently, thus highlighting the need to consider both cognitive load and trust as distinct but interrelated factors in varying the robot autonomy level in shared-control settings. This insight is crucial for the development of more effective and adaptable assistive robotic systems.