We have studied magnetic Feshbach resonances in an ultracold sample of Na prepared in the absolute hyperfine ground state. We report on the observation of three $s$-, eight $d$-, and three $g$-wave Feshbach resonances, including a more precise determination of two known $s$-wave resonances, and one $s$-wave resonance at a magnetic field exceeding 200 mT. Using a coupled-channels calculation we have improved the sodium ground-state potentials by taking into account these new experimental data and derived values for the scattering lengths. In addition, a description of the molecular states leading to the Feshbach resonances in terms of the asymptotic-bound-state model is presented.