We used a sample of Kepler candidate planets with orbital periods less than 200 days and radii between 1.5 and 30 Earth radii (R⊕) to determine the typical dynamical spacing of neighboring planets. To derive the intrinsic (i.e., free of observational bias) dynamical spacing of neighboring planets, we generated populations of planetary systems following various dynamical spacing distributions, subjected them to synthetic observations by the Kepler spacecraft, and compared the properties of observed planets in our simulations with actual Kepler detections. We found that, on average, neighboring planets are spaced 21.7 mutual Hill radii apart with a standard deviation of 9.5. This dynamical spacing distribution is consistent with that of adjacent planets in the solar system. To test the packed planetary systems hypothesis, the idea that all planetary systems are dynamically packed or filled to capacity, we determined the fraction of systems that are dynamically packed by performing long-term (108years) numerical simulations. In each simulation, we integrated a system with planets spaced according to our best-fit dynamical spacing distribution but containing an additional planet on an intermediate orbit. The fraction of simulations exhibiting signs of instability provides an approximate lower bound on the fraction of systems that are dynamically packed; we found that ≥31%, ≥35%, and ≥45% of two-planet, three-planet, and four-planet systems are dynamically packed, respectively. Such sizeable fractions suggest that many planetary systems are indeed filled to capacity. This feature of planetary systems is another profound constraint that formation and evolution models must satisfy. © 2013. The American Astronomical Society. All rights reserved..