Sort of. Just like ni...ni... means neither...nor; and soit...soit... means either...or...
What would this me if it were just, "Viens vite que possible." Or is this no longer grammatically correct if I understand correctly.
"Viens vite que possible." isn't correct French. "aussi ... que ..." implies a comparison. Here we compare the maximum speed of the person to the speed we want him to reach. We want him to come as fast as he possibly can.
Another example: "Il est aussi fort que moi." = "He's as strong as me."