We report on an intermediate intensity level in the emission intermittency of single CdSe/ZnS core shell quantum dots, which has been overlooked in previous experiments most likely due to its low quantum efficiency. The intermediate intensity level is observed in CdSe/ZnS quantum dots of large diameter (about 5 nm diameter) and appears to be independent of the general dark state power law dynamics. The dim emission periods are found to be exponentially distributed and thus correspond to similar findings in CdSe/CdS quantum dots, where their existence has been interpreted in terms of the emission of a positively charged trion.