Levy process decomposition and t at infinity

I have a small question it is know that one can decompose levy into 4 elements, 2 of them are continuous, and the are W_t – wiener process, and the second is a drift term: t.
But there is one thing I cant grasp, when t goes to infinity the process also goes to infinity, and it’s just made me think…
I know that maybe I’m oversimplifying, that why I’m here.
Am I looking at it all wrong?