Background: Vaccine-induced SARS-CoV-2-anti-spike antibody (anti-S/RBD) titers are often used as a marker of immune protection and to anticipate the risk of breakthrough infections, although no clear cut-off is available. We describe the incidence of SARS-CoV-2 vaccine breakthrough infections in COVID-19-free personnel of our hospital, according to B- and T-cell immune response elicited one month after mRNA third dose vaccination. Methods: The study included 487 individuals for whom data on anti-S/RBD were available. Neutralizing antibody titers (nAbsT) against the ancestral Whuan SARS-CoV-2, and the BA.1 Omicron variant, and SARS-CoV-2 T-cell specific response were measured in subsets of 197 (40.5%), 159 (32.6%), and 127 (26.1%) individuals, respectively. Results: On a total of 92,063 days of observation, 204 participants (42%) had SARS-CoV-2 infection. No significant differences in the probability of SARS-CoV-2 infection for different levels of anti-S/RBD, nAbsT, Omicron nAbsT, or SARS-CoV-2 T cell specific response, and no protective thresholds for infection were found. Conclusions: Routine testing for vaccine-induced humoral immune response to SARS-CoV-2 is not recommended if measured as parameters of ‘protective immunity’ from SARS-CoV-2 after vaccination. Whether these findings apply to new Omicron-specific bivalent vaccines is going to be evaluated.