Performing time-tagged, time-correlated, single-photon-counting studies on individual colloidal nanocrystal quantum dots (NQDs), the evolution of photoluminescence (PL) intensity-fluctuation behaviors in near-infrared (NIR) emitting type II, InP/CdS core-shell NQDs is investigated as a function of shell thickness. It is observed that Auger recombination and hot-carrier trapping compete in defining the PL intensity-fluctuation behavior for NQDs with thin shells, whereas the role of hot-carrier trapping dominates for NQDs with thick shells. These studies further reveal the distinct ramifications of altering either the excitation fluence or repetition rate. Specifically, an increase in laser pump fluence results in the creation of additional hot-carrier traps. Alternately, higher repetition rates cause a saturation in hot-carrier traps, thus activating Auger-related PL fluctuations. Furthermore, it is shown that Auger recombination of negatively charged excitons is suppressed more strongly than that of positively charged excitons because of the asymmetry in the electron-hole confinement in type II NQDs. Thus, this study provides new understanding of how both NQD structure (shell thickness and carrier-separation characteristics) and excitation conditions can be used to tune the PL stability, with important implications for room-temperature single-photon generation. Specifically, the first non-blinking NQD capable of single-photon emission in the near-infrared spectral regime is described.