DNA encounters topological problems in vivo because of its extended double-helical structure. As a consequence, the semiconservative mechanism of DNA replication leads to the formation of DNA braids or catenanes, which have to be removed for the completion of cell division. To get a better understanding of these structures, we have studied the elastic behavior of two braided nicked DNA molecules using a magnetic trap apparatus. The experimental data let us identify and characterize three regimes of braiding: a slightly twisted regime before the formation of the first crossing, followed by genuine braids which, at large braiding number, buckle to form plectonemes. Two different approaches support and quantify this characterization of the data. First, Monte Carlo (MC) simulations of braided DNAs yield a full description of the molecules' behavior and their buckling transition. Second, modeling the braids as a twisted swing provides a good approximation of the elastic response of the molecules as they are intertwined. Comparisons of the experiments and the MC simulations with this analytical model allow for a measurement of the diameter of the braids and its dependence upon entropic and electrostatic repulsive interactions. The MC simulations allow for an estimate of the effective torsional constant of the braids (at a stretching force F = 2 pN): C(b) approximately 48 nm (as compared with C approximately 100 nm for a single unnicked DNA). Finally, at low salt concentrations and for sufficiently large number of braids, the diameter of the braided molecules is observed to collapse to that of double-stranded DNA. We suggest that this collapse is due to the partial melting and fraying of the two nicked molecules and the subsequent right- or left-handed intertwining of the stretched single strands.