Background/Objectives: The aims of this study were to establish the relationship between hand grip strength (HGS) and sleep disturbances, as well as to correlate HGS with the perception of general health status. Methods: A cross-sectional study was conducted among Physical Therapy students. Participants completed the International Physical Activity Questionnaire–Short Form (IPAQ-SF), the Pittsburgh Sleep Quality Index (PSQI), and the General Health Questionnaire (GHQ-12). HGS was measured using a dynamometer and self-reported anonymously. Results: A total of 145 students participated (58.6% males; mean age: 21.0 ± 3.9 years). The average HGS was 42.4 kg in the dominant hand and 39.2 kg in the non-dominant one. Poor subjective sleep quality was reported by 27.5%; 84.1% slept less than 7 h. GHQ-12 scores indicated that 31.7% may be experiencing emotional distress. HGS was inversely correlated with PSQI scores in both dominant (ρ = –0.211; p = 0.019) and non-dominant hands (ρ = –0.178; p = 0.049). Students with GHQ-12 scores >12 had significantly lower HGS. No significant correlation was found between HGS and physical activity intensity. Conclusions: Lower hand grip strength was correlated with poor sleep quality and higher GHQ-12 scores, independently of physical activity levels. These findings suggest that HGS may serve as a simple and accessible indicator of psychological vulnerability in university students.