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Patterns of diversification in species‐rich clades provide insight into the processes that generate biological diversity. We tested different models of lineage and phenotypic diversification in an exceptional continental radiation, the ovenbird family Furnariidae, using the most complete species‐level phylogenetic hypothesis produced to date for a major avian clade (97% of 293 species). We found that the Furnariidae exhibit nearly constant rates of lineage accumulation but show evidence of constrained morphological evolution. This pattern of sustained high rates of speciation despite limitations on phenotypic evolution contrasts with the results of most previous studies of evolutionary radiations, which have found a pattern of decelerating diversity‐dependent lineage accumulation coupled with decelerating or constrained phenotypic evolution. Our results suggest that lineage accumulation in tropical continental …