Although randomised trials are widely accepted as the ideal way of obtaining unbiased estimates of treatment effects, some treatments have dramatic effects that are highly unlikely to reflect inadequately controlled biases. We compiled a list of historical examples of such effects and identified the features of convincing inferences about treatment effects from sources other than randomised trials. A unifying principle is the size of the treatment effect (signal) relative to the expected prognosis (noise) of the condition. A treatment effect is inferred most confidently when the signal to noise ratio is large and its timing is rapid compared with the natural course of the condition. For the examples we considered in detail the rate ratio often exceeds 10 and thus is highly unlikely to reflect bias or factors other than a treatment effect. This model may help to reduce controversy about evidence for treatments whose effects are so dramatic that randomised trials are unnecessary.