A method is introduced to estimate the proximity of climate sub-systems to non-linear thresholds. We suggest to measure the smallest decay rate of the system under investigation and to consider its trend. We argue that this is the diagnostic variable most directly linked to the distance from a bifurcation threshold. With the climate model of intermediate complexity CLIMBER2 we demonstrate our method for the North Atlantic thermohaline circulation. It is shown that proper analysis of paleo information could significantly reduce the uncertainty which plagues current estimates of the distance from the shutdown of the thermohaline circulation.