In response to the COVID-19 pandemic, a variety of social distancing measures to mitigate the virus outbreak have been implemented. These measures may have unintended consequences on individuals’ well-being, such as increased stress, anxiety, and sleep disruptions. We investigated the extent to which individuals’ mental health status is associated with perceived changes in sleep amount and sleep quality among a sample of adult twin pairs (N = 909 pairs; 77% MZ, 23% DZ), less than a month after the outbreak was declared a pandemic by the World Health Organization. About half of participants reported no change in sleep amount (50.1%) or sleep quality (55.6%). Approximately one-third of the participants had increased amount of sleep (29.8%), and 32.9% reported a decrease in sleep quality. We found that stress and anxiety levels were associated with sleep reduction (ORs = 2.36 and 3.12 for stress and anxiety, respectively) and poorer sleep quality (ORs = 2.45 and 3.73 for stress and anxiety, respectively), even after taking into account between-family confounds. A much smaller association was observed between levels of stress and anxiety and increased sleep amount (ORs = 1.42 and 1.60 for stress and anxiety, respectively) and sleep quality (OR = 1.21 and 1.29 for stress and anxiety, respectively), which was no longer significant after controlling for between-family confounds. Our results demonstrate that stress and anxiety associated with the COVID-19 pandemic and social distancing measures may be linked to reduced sleep amount and quality.