There is a pressing need for strategies to slow or treat the progression of functional decline in people living with HIV. This paper explores a novel rehabilitation robotics approach to measuring cognitive and motor impairment in adults living with HIV, including a subset with stroke. We conducted a cross-sectional study with 21 subjects exhibiting varying levels of cognitive and motor impairment. We developed three robot-based tasks - trajectory tracking, N-back, and spatial span - to assess if metrics derived from these tasks were sensitive to differences in subjects with varying levels of executive function and upper limb motor impairments. We also examined if these metrics could estimate clinical cognitive and motor scores. The results showed that the average sequence length on the robot-based spatial span task was the most sensitive to differences between subjects' cognitive and motor impairment levels. We observed strong correlations between robot-based measures and clinical cognitive and motor assessments relevant to the HIV population, such as the Color Trails 1 (rho = 0.83), Color Trails 2 (rho = 0.71), Digit Symbol - Coding (rho = 0.81), Montreal Cognitive Assessment - Executive Function subscore (rho = 0.70), and Box and Block Test (rho = 0.74). Importantly, our results highlight that gross motor impairment may be overlooked in the assessment of HIV-related disability. This study shows that rehabilitation robotics can be expanded to new populations beyond stroke, namely to people living with HIV and those with cognitive impairments.