  Is it possible for a large sequence of measurements or observations, which
support a hypothesis, to counterintuitively decrease our confidence? Can
unanimous support be too good to be true? The assumption of independence is
often made in good faith, however rarely is consideration given to whether a
systemic failure has occurred.
  Taking this into account can cause certainty in a hypothesis to decrease as
the evidence for it becomes apparently stronger. We perform a probabilistic
Bayesian analysis of this effect with examples based on (i) archaeological
evidence, (ii) weighing of legal evidence, and (iii) cryptographic primality
testing.
  We find that even with surprisingly low systemic failure rates high
confidence is very difficult to achieve and in particular we find that certain
analyses of cryptographically-important numerical tests are highly optimistic,
underestimating their false-negative rate by as much as a factor of $2^{80}$.
