A small success can lead to more success.  When you complete small tasks, larger ones may follow.  This may come out of your own ambition, or it may be that whomever you are working with believes that you are ready for more. 

Is there a right way to increase the level of responsibility that you get over time?  When you start small and progress organically, you are giving yourself the best chance to succeed because you are learning as you go, brick by brick.