Internet addiction is a serious problem among young adults that requires increased attention, especially at a time of distance learning during the coronavirus disease 2019 (COVID-19) pandemic. The aim of the study was to assess the relationships between internet addiction and selected socio-demographic, study-related, and psychological characteristics of college students. Internet addiction was measured using the Internet Addiction Test both overall and in its individual subscales (Salience, Excessive Use, Neglect Work, Anticipation, Lack of Control, and Neglect Social Life). The selected characteristics represented (1) socio-demographic profile (gender, age, residence, family), (2) academic profile (housing during the semester, form of study), and (3) psychological profile (depressive symptoms—the Patient Health Questionnaire, stress—the Perceived Stress Scale, anxiety symptoms—the Generalized Anxiety Disorder). Data collection took place during the first wave of the COVID-19 pandemic in 2020 at Czech and Slovak colleges, with 1,422 students from the Czech Republic and 1,677 students from Slovakia participating in the research. The analytical processes were carried out through descriptive analysis, non-parametric difference analysis, and multiple negative binomial regression. Mild internet addiction was found in 387 (27.2%) Czech and 452 (27.0%) Slovak students. Moderate internet addiction was identified in 49 (3.4%) students from the Czech Republic and in 100 (6.0%) students from Slovakia. Two (0.1%) Czech and three (0.2%) Slovak students reported severe internet addiction. Increased likelihood of internet addiction overall, as well as in most individual subscales, was found particularly among male students and students who lived away from home during the semester. Depressive symptoms and stress could also be considered significant predictors in both countries. These results are important for the development of effective strategies and prevention programs, as Internet addiction may be a serious problem in the future, given the current times. When assessing internet addiction among college students, it would also be appropriate to evaluate the individual internet addiction subscales and their specifics.