SS didn't explicitly say that assists and eliminations are mutually exclusive (edit: until his second post in the thread, but you asked an unanswered question later on that gave evidence to suggest his second post wasn't entirely correct), so I'm inclined to go with Qbicle's answer that one player can score an elimination and the player who did the second most amount of damage can still score an assist.
The way I see it, "the person who does the most damage either gets an elimination or an assist" does not mean that "if someone gets an elimination no one gets an assist" but rather "if someone gets an elimination the player who did the second most damage gets an assist."