At each point on a 'line', the direction of the 'line' is the direction of the force on a small test magnet placed in the field at that point.
If two 'lines' crossed at the same point, that means a small test magnet placed at that point in the field would feel a force in two different directions.
But even if that were true, then the net effect on the small test magnet would be the vector sum of the two forces, and they would be represented by a single net force anyway, and therefore by a single field 'line' at that point.
