When using amphiphilic polymers to exfoliate and disperse carbon nanotubes in water, the balance between the hydrophobic and hydrophilic moieties is critical and nontrivial. Here, we investigate the mode of surface attachment of a triblock copolymer, Pluronics F127, composed of a central hydrophobic polypropylene oxide block flanked by hydrophilic polyethylene oxide blocks, onto single-walled carbon nanotubes (SWNTs). Crucially, we analyze the composition in dispersant of both the as-obtained dispersion (the supernatant) and the precipitate-containing undispersed materials. For this, we combine the carefully obtained data from 1H NMR peak intensities and self-diffusion and thermogravimetric analysis. The molecular motions behind the observed NMR features are clarified. We find that the hydrophobic blocks attach to the dispersed SWNT surface and remain significantly immobilized leading to 1H NMR signal loss. On the other hand, the hydrophilic blocks remain highly mobile and thus readily detectable by NMR. The dispersant is shown to possess significant block polydispersity that has a large effect on dispersibility. Polymers with large hydrophobic blocks adsorb on the surface of the carbonaceous particles that precipitate, indicating that although a larger hydrophobic block is good for enhancing adsorption, it may be less effective in dispersing the tubes. A model is also proposed that consistently explains our observations in SWNT dispersions and some contradicting findings obtained previously in carbon nanohorn dispersions. Overall, our findings help elucidating the molecular picture of the dispersion process for SWNTs and are of interest when looking for more effective (i.e., well-balanced) polymeric dispersants.