Could you the give us the context in which you heard or read this phrase? All I can think of is something like "His wife died partway through the football game, but he was so engrossed in the game he didn't even notice." ![Sad [:(]](/emoticons/emotion-6.gif)
In other words, one could die "partway through" an event (during; between the beginning and the end). I suppose in a certain context the phrase could mean "partially die" -- "After being hit by a car, the big oak tree died partway through; the side near the street was dead but the other side was still growing."