Multi-part polylines are often the result of data capture. 

Example 1

A classic situation is where digitising has captured the boundary of a lake along with the inflow and outflow.  The resulting feature is a multi-part polyline.  The image below shows a multi-part polyline "exploded" into its individual parts, as you can see the middle part (2) forms a loop.

Multi-part example 1

Part 2 in the image above still needs to be converted to a single centre line for the lake to ensure correct network connectivity.  If the image above was a river braid instead of a lake boundary; the wrongly digitised multi-part polyline (part 2) would need to be split into 2 lines, left and right channel.  So what you do with a multi-part polyline depends upon what is was capturing.

Example 2

A multi-part feature is shown below. The first image shows the polylines with their direction of flow, one polyline is not showing the expected logical direction of flow

Multi-part Example 2a

Selecting the polyline shows up an unusual selection.

Multi-part Example 2b

Entering edit mode and selecting edit vertices (Edit vertices)shows the vertices. You could then use the sketch properties (Sketch properties) dialog to explore the order of the vertices. 

Multi-part Example 2c

Click here to learn how to explode multi-part polylines.

This tool generates an error log file called MultiPartPolylines.txt

WarningOnce you have edited your data you must rebuild the node ID values of the attribute table.  You will want to use the newly created node fields for stream ordering.