In the US, the gap between the wages of immigrants at arrival and those of comparable natives has widened substantially over the last decades, and the subsequent speed of convergence has slowed down. These facts lead to a pessimistic view regarding wage assimilation prospects of immigrants. This paper studies an unexplored mechanism that can explain an important portion of these regularities: labor market competition. Because natives and immigrants are imperfect substitutes in production, increasing immigrant inflows exert stronger labor market competition on previous cohorts of immigrants than on natives, contributing to a widening of the wage gap. We quantify the importance of this mechanism by using a simple model that accounts for the main features of the literature on the wage impacts of immigration and wage assimilation. Our simulations suggest that, if competition and compositional effects are accounted for, the wage assimilation profiles of recent immigrant cohorts are, in fact, very similar to those of earlier cohorts.