While the geographic range of a species is a fundamental unit of macroecology and a leading predictor of extinction risk, the evolutionary dynamics of species' ranges remain poorly understood. Based on statistical associations between range size and species age, many studies have claimed support for general models of range evolution in which the area occupied by a species varies predictably over the course of its life. Such claims have been made using both paleontological data and molecular estimates of the age of extant species. However, using a stochastic model, we show that the appearance of trends in range size with species' age can arise even when range sizes have evolved at random through time. This occurs because the samples of species used in existing studies are likely to be biased with respect to range size: for example, only those species that happened to have large or expanding ranges are likely to survive to the present, while extinct species will tend to be those whose ranges, by chance, declined through time. We compared the relationship between the age and range size of species arising under our stochastic model to those observed across 1,269 species of extant birds and mammals and 140 species of extinct Cenozoic marine mollusks. We find that the stochastic model is able to generate the full spectrum of empirical age-area relationships, implying that such trends cannot be simply interpreted as evidence for models of directional range size evolution. Our results therefore challenge the theory that species undergo predictable phases of geographic expansion and contraction through time.