The outbreak of the new Coronavirus (COVID-19) pandemic has prompted investigations on various aspects. This research aims to study the possible correlation between the numbers of swab tests and the trend of confirmed cases of infection, while paying particular attention to the sickness level. The study is carried out in relation to the Italian case, but the result is of more general importance, particularly for countries with limited ICU (intensive care units) availability. The statistical analysis showed that, by increasing the number of tests, the trend of home isolation cases was positive. However, the trend of mild cases admitted to hospitals, intensive case cases, and daily deaths were all negative. The result of the statistical analysis provided the basis for an AI study by ANN. In addition, the results were validated using a multivariate linear regression (MLR) approach. Our main result was to identify a significant statistical effect of a reduction of pressure on the health care system due to an increase in tests. The relevance of this result is not confined to the COVID-19 outbreak, because the high demand of hospitalizations and ICU treatments due to this pandemic has an indirect effect on the possibility of guaranteeing an adequate treatment for other high-fatality diseases, such as, e.g., cardiological and oncological ones. Our results show that swab testing may play a significant role in decreasing stress on the health system. Therefore, this case study is relevant, in particular, for plans to control the pandemic in countries with a limited capacity for admissions to ICU units.