Objectives: Stress, anxiety, and depression among students have many negative health consequences and may predispose students to poor sleep quality; therefore, this research aimed to investigate the perceived stress, anxiety, and depression and their relation to the level of sleep quality among healthcare students. Methods: A cross-sectional study using a validated survey was conducted among Saudi healthcare students from different regions during the period from 26 September 2022 to 30 October 2022. The Pittsburgh Sleep Quality Index (PSQI) was utilized to assess sleep quality. PSPP Statistical Analysis Software version 1.2.0 was used for all statistical analyses. Results: 701 respondents participated in this research; the response rate was 73.8%. About 60% of the study sample was female, and the average age was 20.9 years. 80.3% of students have poor sleep quality; the participants’ mean sleeping hours per night was 6.81 ± 1.88 h. About three-quarters of students (72.9%) perceive themselves as stressed. A significant positive correlation was observed between sleep quality and perceived stress (r-value = 0.363), anxiety (r-value = 0.387), and depression (r-value = 0.347). Poor sleep quality was more likely among those with stress than those without stress (AOR = 1.79; 95% CI 1.07, 2.99) and two times more likely among students with cases of anxiety than those without anxiety (AOR = 2.07; 95% CI 1.10, 3.94). Conclusions: Our study highlights the high prevalence of poor sleep quality, anxiety, depression, and stress among healthcare students in Saudi Arabia. Students’ reported stress, anxiety, and residence region were associated with poor sleep quality. These results imply the necessity of regular screening and appropriate intervention for sleep problems, stressors, and anxiety among healthcare students.