Basic principles of statistical mechanics require that proteins sample an ensemble of conformations at any nonzero temperature. However, it is still common to treat the crystallographic structure of a protein as the structure of its native state, largely because high-resolution structural characterization of protein flexibility remains a profound challenge. To assess the typical degree of conformational heterogeneity within folded proteins, we construct Markov state models describing the thermodynamics and kinetics of proteins ranging from 72 to 263 residues in length. Each of these models is built from hundreds of microseconds of atomically detailed molecular dynamics simulations. Examination of the side-chain degrees of freedom reveals that almost every residue visits at least two rotameric states over this time frame, with rotamer transition rates spanning a wide range of time scales (from nanoseconds to tens of microseconds). We also report substantial backbone dynamics on time scales longer than are typically addressed by experimental measures of protein flexibility, such as NMR order parameters. Finally, we demonstrate that these extensive rearrangements are consistent with NMR and crystallographic data, which supports the validity of our models. Altogether, these results depict the interior of proteins not as well-ordered solids, as is often imagined, but instead as dense fluids, which undergo substantial structural fluctuations despite their high packing fraction.