Poor mental health is a growing concern among young people during the coronavirus disease 2019 (COVID-19) pandemic. The aim of this study was to assess the associations of Internet addiction with depressive symptoms, anxiety symptoms, and stress in higher education students during the COVID-19 pandemic, as well as to examine these mental health problems in the context of study-related characteristics. The research sample consisted of 3,099 participants from the Czech Republic (CZ: 1,422) and Slovak Republic (SK: 1,677). The Internet Addiction Test (IAT), the Generalized Anxiety Disorder (GAD-7) scale, the Patient Health Questionnaire for depressive symptoms (PHQ-9), and the Perceived Stress Scale (PSS) were used to measure mental health problems. The analyses also included demographic data (gender and age) and study-related characteristics (form of study, degree of study, field of study, distance between college and home, and housing during the semester). Based on the results of frequency and descriptive analyses, the prevalence of mental health problems was high. The most serious levels of Internet addiction (IAT cut-off point ≥ 50), to which attention should be paid, were found in 3.5% of Czech and 6.2% of Slovak students. Using the standard cut-off point of GAD-7 ≥ 10, 14.1% of Czech and 11.6% of Slovak students were identified with anxiety symptoms. Regarding the PHQ-9 with the cut-off point ≥ 10, 23.4% of Czech and 19.1% of Slovak students had depressive symptoms, which should be addressed. Using the PSS cut-off point ≥ 27, 12.9% of Czech students and 9.1% of Slovak students perceived high stress. The quantile regression analysis showed that Internet addiction was positively associated with anxiety symptoms, depressive symptoms, and stress in all of the analyzed cases (p-value < 0.001). In terms of study-related characteristics, the binomial logistic regression analysis revealed that risk factors for mental health problems in Czech and Slovak students were mainly full-time form of study and living away from home during the semester. Internet addiction, anxiety symptoms, depressive symptoms, and stress are issues that require increased attention, and professionals and policy-makers should implement interventions to effectively prevent and help students with psychological problems.