How many network engineers do you think start out as network engineers? Not too many.
Most start out as techs or help desk reps and then work their way up. A CS degree would be most helpful, though not essential, for this path.
Computer Engineering is actually designing circuits and the nitty gritty things that make computers work.
Network Engineers take the technologies that CE's develop, and make it hook together.
Think of CE's as being on the micro level, and NE's being on the macro level.
CE's work on minute details, while NE's network/administrate corporate wide systems.