Rapid and comprehensive social change is required to mitigate pressing environmental issues such as climate change. Social tipping interventions have been proposed as a policy tool for creating this kind of change. Social tipping means that a small minority committed to a target behaviour can create a self-reinforcing dynamic, which establishes the target behaviour as a social norm. The possibility of achieving the large-scale diffusion of pro-environmental norms and related behaviours with an intervention delimited in size and time is tempting. Yet, the canonical model of tipping, the coordination game, may evoke overly optimistic expectations regarding the potential of tipping, due to the underlying assumption of homogenous preferences. Relaxing this assumption, we devise a threshold model of tipping pro-environmental norm diffusion. The model suggests that depending on the distribution of social preferences in a population, and the individual cost of adopting a given pro-environmental behaviour, the same intervention can activate tipping, have little effect, or produce a backlash. Favourable to tip pro-environmental norms are widespread advantageous inequity aversion and low adoption costs. Adverse are widespread self-regarding preferences or disadvantageous inequity aversion, and high costs. We discuss the policy implications of these findings and suggest suitable intervention strategies for different contexts.