Debundling and dispersing single-walled carbon nanotubes (SWNTs) is essential for applications but the process is not well understood. In this work, aqueous SWNT dispersions were produced by sonicating pristine SWNT powder in the presence of an amphiphilic triblock copolymer (Pluronic F127) as dispersant. Upon centrifugation, one obtains a supernatant with suspended individual tubes and thin bundles and a precipitate with large bundles (and impurities). In the supernatant, that constitutes the final dispersion, we determined the dispersed SWNT concentration by thermogravimetric analysis (TGA) and UV-Vis spectroscopy and the dispersant concentration by 1H NMR. The fraction of dispersant adsorbed at the SWNT surface was obtained by 1H diffusion NMR. Sigmoidal dispersion curves recording the concentration of dispersed SWNTs as a function of supernatant dispersant concentration were obtained at different SWNT loadings and sonication times. As SWNT bundles are debundled into smaller and smaller ones, the essential role of the dispersant is to sufficiently quickly cover the freshly exposed surfaces created by shear forces induced during sonication. Primarily kinetic reasons are behind the need for dispersant concentrations required to reach a substantial SWNT concentration. Centrifugation sets the size-threshold below which SWNT particles are retained in the dispersion, and consequently determines the SWNT concentration as a function of sonication time.