This paper provides the first detailed temporal characterization of a multi-wavelength-Brillouinerbium fiber laser (MWBEFL) by measuring the optical intensity of the individual frequency channels with high temporal resolution. It is found that the power in each channel is highly unstable due to the excitation of several cavity modes for typical conditions of operation. Also provided is the real-time measurements of the MWBEFL output power for two configurations that were previously reported to emit phase-locked picosecond pulse trains, concluded from their autocorrelation measurements. Real-time measurements reveal a high degree of instability without the formation of a stable pulse train. Finally, we model the MWBEFL using coupled wave equations describing the evolution of the Brillouin pump, Stokes and acoustic waves in the presence of stimulated Brillouin scattering, and the optical Kerr effect. A good qualitative consistency between the simulation and experimental results is evident, in which the interference signal at the output shows strong instability as well as the chaotic behavior due to the dynamics of participating pump and Stokes waves.