This is the case in Louisiana, where the Atchafalaya River splits from the Mississippi River. In some cases, a distributary can "steal" so much water from its parent that it can become the main route. This is actually a serious problem for New Orleans, because the Atchafalaya takes a steeper route to the Gulf of Mexico, and therefore takes more and more water from the Mississippi. This was recently stopped by the construction of a dam to control the outflow from the Mississippi into the Atchafalaya.
The opposite of a distributary is a tributary