Mass vaccination campaigns against SARS-CoV-2 are under way in many countries with the hope that increasing vaccination coverage will enable reducing current physical distancing measures. Compliance with these measures is waning, while more transmissible virus variants such as B.1.1.7 have emerged. Using SARS-CoV-2 transmission model we investigated the impact of the feedback between compliance, the incidence of infection, and vaccination coverage on the success of a vaccination programme in the population where waning of compliance depends on vaccine coverage. Our results suggest that the combination of fast waning compliance, slow vaccination rates, and more transmissible variants may result in a higher cumulative number of infections than in a situation without vaccination. These adverse effects can be alleviated if vaccinated individuals do not revert to pre-pandemic contact rates, and if non-vaccinated individuals remain compliant with physical distancing measures. Both require convincing, clear and appropriately targeted communication strategies by public health authorities.