At a single-molecule level, fluorophore emission intensity fluctuates between bright and dark states. These fluctuations, known as blinking, limit the use of fluorophores in single-molecule experiments. The dark-state duration shows a universal heavy-tailed power-law distribution characterized by the occurrence of long non-emissive periods. Here we have synthesized novel CdSe-CdS core-shell quantum dots with thick crystalline shells, 68% of which do not blink when observed individually at 33 Hz for 5 min. We have established a direct correlation between shell thickness and blinking occurrences. Importantly, the statistics of dark periods that appear at high acquisition rates (1 kHz) are not heavy tailed, in striking contrast with previous observations. Blinking statistics are thus not as universal as thought so far. We anticipate that our results will help to better understand the physico-chemistry of single-fluorophore emission and rationalize the design of other fluorophores that do not blink.