A major goal of research in ecology and evolution is to explain why species richness varies across habitats, regions, and clades. Recent reviews have argued that species richness patterns among regions and clades may be explained by "ecological limits" on diversity over time, which are said to offer an alternative explanation to those invoking speciation and extinction (diversification) and time. Further, it has been proposed that this hypothesis is best supported by failure to find a positive relationship between time (e.g., clade age) and species richness. Here, I critically review the evidence for these claims, and propose how we might better study the ecological and evolutionary origins of species richness patterns. In fact, ecological limits can only influence species richness in clades by influencing speciation and extinction, and so this new "alternative paradigm" is simply one facet of the traditional idea that ecology influences diversification. The only direct evidence for strict ecological limits on richness (i.e., constant diversity over time) is from the fossil record, but many studies cited as supporting this pattern do not, and there is evidence for increasing richness over time. Negative evidence for a relationship between clade age and richness among extant clades is not positive evidence for constant diversity over time, and many recent analyses finding no age-diversity relationship were biased to reach this conclusion. More comprehensive analyses strongly support a positive age-richness relationship. There is abundant evidence that both time and ecological influences on diversification rates are important drivers of both large-scale and small-scale species richness patterns. The major challenge for future studies is to understand the ecological and evolutionary mechanisms underpinning the relationships between time, dispersal, diversification, and species richness patterns.