Except in very rare configurations (i.e. not 99.9% of residential), you do not want to have multiple paths to ground within a system. All grounds should go to the tied ground/neutral bus in the main breaker panel, which then goes to earth via a ground rod or a clamp to a copper gas/water line, etc. Otherwise you can have current flowing in ways that the system isn’t designed for, which at the least can trip breakers and GFCIs, and at worst exceed the rating of the wires in a short condition and cause a fire.