For long-term deployment in dynamic real-world environments, assistive robots must continue to learn and adapt to their environments. Researchers have developed various computational models for continual learning (CL) that can allow robots to continually learn from limited training data, and avoid forgetting previous knowledge. While these CL models can mitigate forgetting on static, systematically collected datasets, it is unclear how human users might perceive a robot that continually learns over multiple interactions with them. In this paper, we developed a system that integrates CL models for object recognition with a Fetch mobile manipulator robot and allows human participants to directly teach and test the robot over multiple sessions. We conducted an in-person study with 60 participants that interacted with our system in 300 sessions (5 sessions per participant). We conducted a between-subject study with three different CL models to understand human perceptions of continual learning robots over multiple sessions. Our results suggest that participants' perceptions of trust, competence, and usability of a continual learning robot significantly decrease over multiple sessions if the robot forgets previously learned objects. However, the perceived task load on participants for teaching and testing the robot remains the same over multiple sessions even if the robot forgets previously learned objects. Our results also indicate that state-of-the-art CL models might perform unreliably when applied on robots interacting with human participants. Further, continual learning robots are not perceived as very trustworthy or competent by human participants, regardless of the underlying continual learning model or the session number.