Background/Objectives: Mental disorders in nursing students, although very common under normal circumstances, have worsened over time due to the COVID-19 pandemic. This study aimed to assess (1) what the prevalence of anxiety and depression symptoms in Italian nursing students was and (2) what factors were associated with them. Methods: In May 2023, a cross-sectional study was conducted on the bachelor’s degree Nursing course in an Italian university. Levels of anxiety and stress were assessed using the Generalized Anxiety Disorder Scale (GAD-7) and the Patient Health Questionnaire (PHQ-9), respectively. In addition, socio-demographic variables and data on mental health, drug intake, and substance use were collected. Multiple binary logistic regression adjusted analyses were used to identify predictive factors. Results: A total of 148 nursing students completed the questionnaire. A total of 9.5% reported a previous diagnosis of a mental disorder, and, of these, 35.7% reported that it was diagnosed after the start of the COVID-19 pandemic. The mean GAD-7 and PHQ-9 score was 9.68 (SD = 5.2) and 8.37 (SD = 5.6), respectively. Protective factors for depressive symptoms included not having a previous diagnosis of a mental disorder (adjusted odds ratio = 0.10, 95% CI: 0.002–0.47, p < 0.001) and a perceived high (adjusted odds ratio = 0.03, 95% CI: 0.003–0.22 p < 0.001) or medium (adjusted odds ratio = 0.14, 95% CI: 0.03–0.82, p < 0.05) socioeconomic level, while the predictor of anxiety symptoms was returning home once a month for students studying away from their residence (adjusted odds ratio = 6.4, 95% CI: 1.01–40.8, p < 0.05). Conclusions: Urgent investments are needed in universities to implement mental health promotion programmes and to offer counselling services to reduce and prevent mental health problems among students.