We report our study of the evolution of superconductivity and the phase diagram of the ternary Fe(Se1-xTex)0.82 (0<=x<=1.0) system. We discovered a new superconducting phase with Tc,max = 14 K in the 0.3 < x < 1.0 range. This superconducting phase is suppressed when the sample composition approaches the end member FeTe0.82, which exhibits an incommensurate antiferromagnetic order. We discuss the relationship between the superconductivity and magnetism of this material system in terms of recent results from neutron scattering measurements. Our results and analyses suggest that superconductivity in this new class of Fe-based compounds is associated with magnetic fluctuations, and therefore may be unconventional in nature.