Semiconducting single-walled carbon nanotubes are one-dimensional materials with great prospects for applications such as optoelectronic and quantum information devices. Yet, their optical performance is hindered by low fluorescent yield. Highly mobile excitons interacting with quenching sites are attributed to be one of the main non-radiative decay mechanisms that shortens the exciton lifetime. In this paper we report on time-integrated photoluminescence measurements on individual polymer wrapped semiconducting carbon nanotubes. An ultra narrow linewidth we observed demonstrates intrinsic exciton dynamics. Furthermore, we identify a state filling effect in individual carbon nanotubes at cryogenic temperatures as previously observed in quantum dots. We propose that each of the CNTs is segmented into a chain of zero-dimensional states confined by a varying local potential along the CNT, determined by local environmental factors such as the amount of polymer wrapping. Spectral diffusion is also observed, which is consistent with the tunneling of excitons between these confined states.