Incomplete lineage sorting (ILS) and introgression increase genealogical discordance across the genome, which complicates phylogenetic inference. In such cases, identifying orthologs that result in gene trees with low estimation error is crucial because phylogenomic methods rely on accurate gene histories. We sequenced whole genomes for the tinamous (Aves: Tinamidae) to dissect the sources of gene and species-tree discordance and reconstruct their interrelationships. We compared results based on four ortholog sets: (1) coding genes (BUSCOs), (2) ultraconserved elements (UCEs) with short flanking regions, (3) UCEs with intermediate flanks, and (4) UCEs with long flanks. We hypothesized that orthologs with more phylogenetically informative sites would result in more accurate species trees because the resulting gene trees contain lower error. Consistent with our hypothesis, we found that long UCEs had the most informative sites and lowest rates of error. However, despite having many informative sites, BUSCO gene trees contained high error compared to long UCEs. Unlike UCEs, BUSCO gene sequences showed a positive association between the proportion of parsimony informative sites and gene tree error. Thus, BUSCO and UCE datasets have different underlying properties of molecular evolution, and these differences should be considered when selecting loci for phylogenomic analysis. Still, species trees from different datasets were mostly congruent. Only one clade, with a history of ILS and introgression, exhibited substantial species-tree discordance across the different data sets. Overall, we present the most complete phylogeny for tinamous to date, identify a new species, and provide a case study for species-level phylogenomic analysis using whole-genomes.