This research has two main goals. First, we present the atmospheric structure and the fundamental parameters of three red super-giants (RSGs), increasing the sample of RSGs observed by near-infrared spectro-interferometry. We carried out spectro-interferometric observations of the RSGs V602 Car, HD 95687, and HD 183589 in the near-infrared K-band (1.92-2.47 mum) with the VLTI/AMBER instrument at medium spectral resolution (R ~ 1500). To categorize and comprehend the extended atmospheres, we compared our observational results to predictions by available hydrostatic PHOENIX, available 3D convection, and new 1D self-excited pulsation models of RSGs. The location of our RSG sources in the Hertzsprung-Russell diagram is confirmed to be consistent with the red limits of recent evolutionary tracks. The observed extensions of the atmospheric layers of our sample of RSGs are comparable to those of Mira stars. This confirms that neither convection nor pulsation alone can levitate the molecular atmospheres of RSGs. Our observed correlation of atmospheric extension with luminosity supports a scenario of radiative acceleration on Doppler-shifted molecular lines.