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Temporal trends in biological invasions are often described by a lag‐phase of little or no increase in species occurrence followed by an increase‐phase in which species occurrence rises rapidly. While several biological and environmental mechanisms may underlie lag‐phases, they may also represent statistical artefacts or temporal changes in sampling effort. To date, distinguishing the facts from these artefacts has not been possible. Here we describe a method for estimating the lag‐phase in cumulative records of species occurrence, using a piecewise regression model that explicitly differentiates the lag and increase phases. We used the von Bertalanffy, logistic, linear and exponential functions to model the increase phase, and identified the best‐fitting function using model selection techniques. We confirmed the accuracy of our method using simulated data and then estimated the length of the lag‐phase (t lag ), the maximum collection rate (r) and the projected asymptotic number of records (K) using herbarium records for 105 weed species in New Zealand, while accounting for changes in sampling effort. Nearly all the New Zealand weed species had a lag‐phase, which averaged around 20–30 years, with 4% of species having a lag‐phase greater than 40 years. In more than two thirds of the cases, the accumulation of records was best modelled with the decelerating von Bertalanffy function, despite the tendency for temporal variation in sampling effort to force cumulative herbarium records to follow the sigmoidal shape of a logistic curve. A positive correlation between r and K is consistent with the assumption that the final distribution of an alien plant species reflects its rate of spread. Seemingly rare but fast‐spreading aliens may thus become tomorrow's noxious weeds. A positive correlation between inflection year and r warns that the weeds that have only begun to spread relatively recently may spread faster than previously known invaders.