The search for traits that make alien species invasive has mostly concentrated on comparing successful invaders and different comparison groups with respect to average trait values. By contrast, little attention has been paid to trait variability among invaders. Here, we combine an analysis of trait differences between invasive and non-invasive species with a comparison of multidimensional trait variability within these two species groups. We collected data on biological and distributional traits for 1402 species of the native, non-woody vascular plant flora of Austria. We then compared the subsets of species recorded and not recorded as invasive aliens anywhere in the world, respectively, first, with respect to the sampled traits using univariate and multiple regression models; and, second, with respect to their multidimensional trait diversity by calculating functional richness and dispersion metrics. Attributes related to competitiveness (strategy type, nitrogen indicator value), habitat use (agricultural and ruderal habitats, occurrence under the montane belt), and propagule pressure (frequency) were most closely associated with invasiveness. However, even the best multiple model, including interactions, only explained a moderate fraction of the differences in invasive success. In addition, multidimensional variability in trait space was even larger among invasive than among non-invasive species. This pronounced variability suggests that invasive success has a considerable idiosyncratic component and is probably highly context specific. We conclude that basing risk assessment protocols on species trait profiles will probably face hardly reducible uncertainties.