We report on a time-resolved spectroscopy of the 63 brightest bursts of SGR J1550-5418, detected with the Fermi/Gamma-ray Burst Monitor during its 2008-2009 intense bursting episode. We performed spectral analysis down to 4 ms timescales to characterize the spectral evolution of the bursts. Using a Comptonized model, we find that the peak energy, E peak, anti-correlates with flux, while the low-energy photon index remains constant at -0.8 up to a flux limit F 10-5 erg s-1 cm-2. Above this flux value, the E peak-flux correlation changes sign, and the index positively correlates with the flux reaching 1 at the highest fluxes. Using a two blackbody model, we find that the areas and fluxes of the two emitting regions correlate positively. Further, we study here for the first time the evolution of the temperatures and areas as a function of flux. We find that the area-kT relation follows the lines of constant luminosity at the lowest fluxes, R 2kT -4, with a break at the higher fluxes (F > 10-5.5 erg s-1 cm-2). The area of the high-kT component increases with the flux while its temperature decreases, which we interpret as being due to an adiabatic cooling process. The area of the low-kT component, on the other hand, appears to saturate at the highest fluxes, toward R max 30 km. Assuming that crust quakes are responsible for soft gamma repeater (SGR) bursts and considering R max as the maximum radius of the emitting photon-pair plasma fireball, we relate this saturation radius to a minimum excitation radius of the magnetosphere, and we put a lower limit on the internal magnetic field of SGR J1550-5418, B int ≳ 4.5 × 1015 G.