Yes, it’s possible to get COVID-19 even if you’ve been vaccinated. No vaccines are 100% effective. In fact, breakthrough cases (when someone tests positive more than two weeks after they're fully vaccinated) are expected, especially as the SARS-CoV-2 virus mutates.
The vaccines significantly reduce — but don’t eliminate — your risk of infection. The risk of a severe illness or death from a breakthrough infection is very low.