DNA bridging can be used to induce specific attractions between small particles, providing a highly versatile approach to creating unique particle-based materials having a variety of periodic structures. Surprisingly, given the fact that the thermodynamics of DNA strands in solution are completely understood, existing models for DNA-induced particle interactions are typically in error by more than an order of magnitude in strength and a factor of two in their temperature dependence. This discrepancy has stymied efforts to design the complex temperature, sequence and time-dependent interactions needed for the most interesting applications, such as materials having highly complex or multicomponent microstructures or the ability to reconfigure or self-replicate. Here we report high-spatial resolution measurements of DNA-induced interactions between pairs of polystyrene microspheres at binding strengths comparable to those used in self-assembly experiments, up to 6 k(B)T. We also describe a conceptually straightforward and numerically tractable model that quantitatively captures the separation dependence and temperature-dependent strength of these DNA-induced interactions, without empirical corrections. This model was equally successful when describing the more complex and practically relevant case of grafted DNA brushes with self-interactions that compete with interparticle bridge formation. Together, our findings motivate a nanomaterial design approach where unique functional structures can be found computationally and then reliably realized in experiment.