Not going to be a ground problem, 99% of the time if one of the filiments light up, then there is enough ground for the other one too.
It sounds to me like you have the wrong bulb in your tail light. Remove the lens, pop out the bulb, and look at how many filiments are in there...there should be two, it sounds to me like you have a back up lamp in there...or maybe the correct one that is shorted out internally.
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