Why do the field lines of an electric or magnetic field never cross?
Field lines represent the net force placed on a charge. Crossed field lines don't occur for the reason outlined below.
Electric field lines are a representation of the direction of the force that will act on a positive charge placed at the point.
Think for a moment about what crossing field lines would mean: that a test charge placed at that point could be experiencing a force in two directions at once. If that was the case we could add the force vectors to find the resultant force - and we'd be back to a single uncrossed field line!