Studies of emergent constraints have frequently proposed that a single metric can constrain future responses of the Earth system to anthropogenic emissions. Here, we illustrate that strong relationships between observables and future climate across an ensemble can arise from common structural model assumptions with few degrees of freedom. Such cases have the potential to produce strong yet overconfident constraints when processes are represented in a common, oversimplified fashion throughout the ensemble. We consider these issues in the context of a collection of published constraints and argue that although emergent constraints are potentially powerful tools for understanding ensemble response variation and relevant observables, their naïve application to reduce uncertainties in unknown climate responses could lead to bias and overconfidence in constrained projections. The prevalence of this thinking has led to literature in which statements are made on the probability bounds of key climate variables that were confident yet inconsistent between studies. Together with statistical robustness and a mechanism, assessments of climate responses must include multiple lines of evidence to identify biases that can arise from shared, oversimplified modelling assumptions that impact both present and future climate simulations in order to mitigate against the influence of shared structural biases.