In this paper, we show that the wage assimilation of immigrants is the result of the intricate interplay between individual skill accumulation and dynamic equilibrium effects in the labor market. When immigrants and natives are imperfect substitutes, increasing immigrant inflows widen the wage gap between them. Using a simple production function framework, we show that this labor market competition channel can explain about one quarter of the large increase in the average immigrant-native wage gap in the United States between the 1960s and 1990s arrival cohorts. Once competition effects and compositional changes in education and region of origin are accounted for, we find that the unobservable skills of newly arriving immigrants increased over time rather than decreased as traditionally argued in the literature. We corroborate this finding by documenting closely matching patterns for immigrants' English language proficiency.