Error: "No route from location 1 to 2"

I want to find the shortest route connecting three points on a map.
I built up a road network, I snapped three points on the road.

I got this error:

Warning: No route from location “Location 1” to location “Location 2”.
Error: No solution found.

I can find the road between point 1 and 3 (of the first map), but I got that error for the road between 3 and 2 and 1 and 2. The road between 1 and 3 does not even look like it is the shortest path (see below).

Is it possible that can depend on some routes feature that does not me allow me to drive through those? so I can’t reach the point 2 and I don’t take
the shortest roads.

The procedure I followed:

1) I snapped the points to the road as described here: https://gis.stackexchange.com/a/1433/73176

2) I created the network from the road polyline similarly to this tutorial: http://desktop.arcgis.com/en/arcmap/latest/extensions/network-analyst/exercise-1-creating-a-network-dataset.htm

3) “Network Analyst” –> “New Route”

4) Right click on “Stops”, “Add locations..” I added the three points

5) “Solve”