We present new 5.2-14.5 μm low-resolution spectra of 14 mid-L to mid-T dwarfs. We also present new 3.0-4.1 μm spectra for five of these dwarfs. These data are supplemented by existing red and near-infrared spectra (∼0.6-2.5 μm), as well as red through mid-infrared spectroscopy of seven other L and T dwarfs presented by Cushing et al. We compare these spectra to those generated from the model atmospheres of Saumon & Marley. The models reproduce the observed spectra well, except in the case of one very red L3.5 dwarf, 2MASS J22244381−0158521. The broad wavelength coverage allows us to constrain almost independently the four parameters used to describe these photospheres in our models: effective temperature (T eff ), surface gravity, grain sedimentation efficiency (f sed ), and vertical gas transport efficiency (K zz ). The CH 4 bands centered at 2.2, 3.3, and 7.65 μm and the CO band at 2.3 μm are sensitive to K zz , and indicates that chemical mixing is important in all L and T dwarf atmospheres. The sample of L3.5 to T5.5 dwarfs spans the range 1800 K T eff 1000 K, with an L-T transition (spectral types L7 to T4) that lies between 1400 and 1100 K for dwarfs with typical near-infrared colors; bluer and redder dwarfs can be 100 K warmer or cooler, respectively, when using infrared spectral types. When using optical spectral types, the bluer dwarfs have more typical T eff values as they tend to have earlier optical spectral types. In this model analysis, f sed increases rapidly between types T0 and T4, indicating that increased sedimentation can explain the rapid disappearance of clouds at this stage of brown dwarf evolution. There is a suggestion that the transition to dust-free atmospheres happens at lower temperatures for lower gravity dwarfs.