The COVID-19 era brought about new medical challenges, which, together with nosocomial bacterial infections, resulted in an enormous burden for the healthcare system. One of the most alarming nosocomial threats was carbapenem-resistant Klebsiella pneumoniae (CRKP). Monitoring CRKP incidence and antimicrobial resistance globally and locally is vitally important. In a retrospective study, the incidence of CRKP in the pre-COVID-19 period (2017–2019) and the COVID-19 pandemic (2020–2022) was investigated in the Central Military Hospital in Ružomberok, Slovak Republic. The relative incidence of CRKP significantly increased during the COVID-19 period—by 4.8 times, from 0.18 to 0.76%. At the same time, 47% of CRKP-positive patients also had COVID-19. Twenty-six KPC and sixty-nine NDM-producing isolates were identified. CRKPs isolated in the year 2022 were submitted to whole genome sequencing, and their susceptibility was tested to cefiderocol, ceftazidime–avibactam, imipenem–relebactam and meropenem–vaborbactam, with excellent results. KPC-producing isolates were also highly susceptible to colistin (92%). The NDM isolates revealed lower susceptibility rates, including only 57% colistin susceptibility. ST-307 prevailed in KPC and ST-11 in NDM isolates. Despite the excellent activity of new antimicrobials, rational antibiotic policy must be thoroughly followed, supported by complementary treatments and strict anti-epidemic precautions.