The clinical trials of the COVID-19 vaccines that are authorized in the European Union have revealed high efficacy in preventing symptomatic infections. However, during vaccination campaigns, some vaccine recipients, including those partially and fully vaccinated, will experience severe COVID-19, requiring hospitalization. This may particularly concern patients with a diminished immune response to the vaccine, as well as non-responders. This work has retrospectively analyzed the 92 cases of patients who were hospitalized between 27 December 2020 and 31 May 2021 in four Polish healthcare units due to COVID-19, and who have previously received the COVID-19 vaccine (54.3% ≤ 14 days after the first dose, 26.1% > 14 days after the first dose, 7.6% ≤ 14 days after the second dose, and 12% > 14 days after the second dose). These patients represented a minute fraction (1.2%) of all the COVID-19 patients who were hospitalized during the same period in the same healthcare institutions. No significant differences in white blood count, absolute lymphocyte count nadir, C-reactive protein, interleukin-6, procalcitonin, oxygen saturation, lung involvement, and fever frequency were found between the recipients of the first and second vaccine dose. A total of 15 deaths were noted (1.1% of all fatal COVID-19 cases in the considered period and healthcare units), including six in patients who received the second dose (five > 14 days after the second dose)—three of these subjects were using immunosuppressive medicines, and two were confirmed to be vaccine non-responders. The study reassures that severe COVID-19 and deaths are not common in vaccinated individuals, highlights that the clinical course in such patients may not reveal any distinctive features, and advocates for close monitoring of those at a higher risk of vaccine failure.