In contrast to previous studies on the political opportunity structures of anti-immigrant parties, this article argues that voters’ perceptions of policy convergence between mainstream alternatives affect their short-term propensity for supporting such partisan challengers. Drawing upon leading research in the field, two hypotheses about voters’ perceptions of policy convergence, in two policy areas (economic redistribution and immigration), are presented. The main findings in the article suggest that policy convergence between mainstream parties has a more immediate impact on the electorate than commonly recognised. Using unique data from Sweden, the article shows that perceived convergence between Swedish mainstream parties in the field of immigration policy increases potential support for the anti-immigrant party, the Sweden Democrats (SD). Yet the results are the opposite when it comes to perceptions of convergence in the field of economic-distributive policies. In contrast to widespread assumptions, the article thus finds that policy convergence between mainstream parties only appears to create short-term opportunities for anti-immigrant parties if it takes place on their own policy turf. These results indicate, in other words, that the potential electorate of the SD – which is a wider group than hard-core xenophobes – is largely driven by preferences about immigration policy, rather than the short-term urge to protest against mainstream parties. The article, therefore, concludes that the cordon sanitaire to isolate the SD in Sweden – which is underpinned by de facto convergence between mainstream parties on immigration policy – could improve, and is unlikely to curb, the short-term electoral opportunities of this party.