We present the detailed rheological changes that occur when small quantities of single-wall carbon nanotubes are dispersed in a poly(acrylic acid) water solution, around the overlap polymer concentration, up to the gel point. Here, pH is used to tune the gel formation. Suspensions of nanotubes at pH <= 5 are exfoliated and dispersed by the polymer. Contacts between the nanotubes are mainly through polymer entangling, and the suspension is viscoelastic. At pH > 5, the polymer is charged, and the solution is not a good solvent for the nanotubes anymore. Nanotube bundles covered with polymer are formed and mechanically percolate along the fluid until they become arrested. As a consequence, the rheological behavior is dominated by a mesoscale superstructure formed by nanotubes and polymer, where viscoelasticity is lost and the suspension becomes elastic. At pH >= 9, the surroundings for the nanotubes are worse, bundles and flocs grow to a larger extent, and they can be observable by scanning microscopies. When the suspension becomes a critical gel, the relaxation moduli can be modeled by a power law in the frequency domain in agreement with the model developed by Winter and co-workers.