We present multi-wavelength observations and modeling of gamma-ray bursts (GRBs) that exhibit a simultaneous re-brightening in their X-ray and optical light curves, and are also detected at radio wavelengths. We show that the re-brightening episodes can be modeled by injection of energy into the blastwave and that in all cases the energy injection rate falls within the theoretical bounds expected for a distribution of energy with ejecta Lorentz factor. Our measured values of the circumburst density, jet opening angle, and beaming-corrected kinetic energy are consistent with the distribution of these parameters for long-duration GRBs at both $z\sim 1$ and $z\gtrsim 6$, suggesting that the jet launching mechanism and environment of these events are similar to that of GRBs that do not have bumps in their light curves. However, events exhibiting re-brightening episodes have lower radiative efficiencies than average, suggesting that a majority of the kinetic energy of the outflow is carried by slow-moving ejecta, which is further supported by steep measured distributions of the ejecta energy as a function of Lorentz factor. We do not find evidence for reverse shocks over the energy injection period, implying that the onset of energy injection is a gentle process. We further show that GRBs exhibiting simultaneous X-ray and optical re-brightenings are likely the tail of a distribution of events with varying rates of energy injection, forming the most extreme events in their class. Future X-ray observations of GRB afterglows with Swift and its successors will thus likely discover several more such events, while radio follow-up and multi-wavelength modeling of similar events will unveil the role of energy injection in GRB afterglows.