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Abstract An increasing number of studies using genetic data at the generic and species levels reveal complex patterns of relationships among populations and species. Incomplete lineage sorting and reticulation are, for instance, commonly observed in low‐rank phylogenies. These two processes falsify the simplifying assumptions often used when reconstructing phylogenetic relationships or when assigning specimens to species using barcodes, i.e., the absence of ancestral polymorphism or the obligate dichotomous relationship among species. We recognize seven processes that act as veils to obscure species boundaries and relationships and lead to erroneous interpretations if not diagnosed and analysed properly. These processes include the transfer of genetic material from plastids or mitochondria to the nuclear genome (NuPt, NuMt), hybridization, the stochasticity of lineage sorting within and among species, genome organization, species demography, selection, genetic and geographic structure. Most of these processes have a direct impact on species effective population sizes, a central parameter to be considered. Failure to take the seven processes into account can affect the accuracy of barcode‐based identification, in particular between closely related species. It can also bias the delimitation of species, result in inaccurate estimations of species relationships and lead to incorrect estimation of species ages. We suggest that these problems can be identified and, to some extent, mitigated by ensuring that the full spatial and genetic ranges of the species are sampled, and by the use of information from all genomes of the species. By analysing numerous loci, each as a separate entity in a coalescence framework, it is possible to take into account stochasticity as well as variation in genome effective population sizes. Such a critical approach would allow the use of genetic data to make realistic assessments of species delimitation, affinities and ages, and should promote a more biologically realistic view of species and speciation.