  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~1 and z>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.
