It is a wide-spread assumption that multiple pieces of evidence, whether they involve measurements or testimony, provide stronger evidential support when they are independent than when they are not. The standard view is that non-independence creates redundancy and leads to over-confidence (see e.g., Soll, 1999; Nisbett & Ross, 1980). In keeping with this, fields as diverse as law, statistics, and philosophy have sought to understand the implications of non-independence and set out rules for dealing with it. In this paper, we set out how the challenges posed by non-independence are both far more wide-spread and more challenging than typically assumed. Specifically, we review how Bayesian probabilistic analysis reveals “simple” inference cases that conflict with wide-spread assumptions about the value of independence. These suggest that mere intuition is not a reliable guide in this arena. However, we then show that the same framework cannot (in its current form) be scaled to common real-world situations even in principle. We conclude with a discussion of the more modest, partial, solutions that are currently available and outline possible future directions.