In Bayesian phylogenetics, the coalescent process provides an informative framework for inferring changes in the effective size of a population from a phylogeny (or tree) of sequences sampled from that population. Popular coalescent inference approaches such as the Bayesian Skyline Plot, Skyride and Skygrid all model these population size changes with a discontinuous, piecewise-constant function but then apply a smoothing prior to ensure that their posterior population size estimates transition gradually with time. These prior distributions implicitly encode extra population size information that is not available from the observed coalescent data i.e., the tree. Here we present a novel statistic, Ω, to quantify and disaggregate the relative contributions of the coalescent data and prior assumptions to the resulting posterior estimate precision. Our statistic also measures the additional mutual information introduced by such priors. Using Ω we show that, because it is surprisingly easy to over-parametrise piecewise-constant population models, common smoothing priors can lead to overconfident and potentially misleading inference, even under robust experimental designs. We propose Ω as a useful tool for detecting when effective population size estimates are overly reliant on prior assumptions and for improving quantification of the uncertainty in those estimates.