(1) Background: Massive social efforts to prevent the spread of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pandemic have affected the epidemiological features of respiratory infections. (2) Methods: The study aims to describe the trend of hospitalizations for bronchiolitis among newborns and infants up to three months of life in Rome (Italy), in the pre-COVID-19 era and during the pandemic. (3) Results: We observed a marked decrease in the number of neonates and infants with bronchiolitis after national lockdowns in 2020 and the first months of 2021 and a similar trend in the number of bronchiolitis caused by respiratory syncytial virus (RSV). RSV was the leading pathogen responsible for bronchiolitis before the national lockdown in March 2020 (70.0% of cases), while Rhinovirus was the leading pathogen responsible for bronchiolitis (62.5%) during the pandemic while strict restrictions were ongoing. As Italy approached the COVID-19 vaccination target, the national government lifted some COVID-19-related restrictions. A surprising rebound of bronchiolitis (particularly cases caused by RSV) was observed in October 2021. (4) Conclusions: In this study, we describe for the first time the fluctuations over time of RSV bronchiolitis among newborns and young infants in Italy in relation to the restrictive measures containing the spread of the COVID-19 pandemic. Our results are in line with other countries’ reports.