We report a systematic study of photoluminescence (PL) intensity and lifetime fluctuations in individual CdSe/CdS core/shell nanocrystal quantum dots (NQDs) as a function of shell thickness. We show that while at low pump intensities PL blinking in thin-shell (4-7 monolayers, MLs) NQDs can be described by random switching between two states of high (ON) and low (OFF) emissivities, it changes to the regime with a continuous distribution of ON intensity levels at high pump powers. A similar behavior is observed in samples with a medium shell thickness (10-12 MLs) without, however, the PL intensity ever switching to a complete "OFF" state and maintaining ca. 30% emissivity ("gray" state). Further, our data indicate that highly stable, blinking-free PL of thick-shell (15-19 MLs) NQDs ("giant" or g-NQDs) is characterized by nearly perfect Poisson statistics, corresponding to a narrow, shot-noise limited PL intensity distribution. Interestingly, in this case the PL lifetime shortens with increasing pump power and the PL decay may deviate from monoexponential. However, the PL intensity distribution remains shot-noise limited, indicating the absence of significant quantum yield fluctuations at a given pump power intensity during the experimental time window.