The closure of the Isthmus of Panama has long been considered to be one of the best defined biogeographic calibration points for molecular divergence-time estimation. However, geological and biological evidence has recently cast doubt on the presumed timing of the initial isthmus closure around 3 Ma but has instead suggested the existence of temporary land bridges as early as the Middle or Late Miocene. The biological evidence supporting these earlier land bridges was based either on only few molecular markers or on concatenation of genome-wide sequence data, an approach that is known to result in potentially misleading branch lengths and divergence times, which could compromise the reliability of this evidence. To allow divergence-time estimation with genomic data using the more appropriate multispecies coalescent (MSC) model, we here develop a new method combining the single-nucleotide polymorphism-based Bayesian species-tree inference of the software SNAPP with a molecular clock model that can be calibrated with fossil or biogeographic constraints. We validate our approach with simulations and use our method to reanalyze genomic data of Neotropical army ants (Dorylinae) that previously supported divergence times of Central and South American populations before the isthmus closure around 3 Ma. Our reanalysis with the MSC model shifts all of these divergence times to ages younger than 3 Ma, suggesting that the older estimates supporting the earlier existence of temporary land bridges were artifacts resulting at least partially from the use of concatenation. We then apply our method to a new restriction-site associated DNA-sequencing data set of Neotropical sea catfishes (Ariidae) and calibrate their species tree with extensive information from the fossil record. We identify a series of divergences between groups of Caribbean and Pacific sea catfishes around 10 Ma, indicating that processes related to the emergence of the isthmus led to vicariant speciation already in the Late Miocene, millions of years before the final isthmus closure.