Mapping Center Answer:

Once you've got leader lines turned on, the size of the gap between the label and the feature will determine whether a leader is shown.  That gap is the distance between the closest point on the feature to the anchor point for the leader on your text.  There are only a few valid anchor points on a text element, and these are at the corners and in the middle of each side. 

So, when you are setting up the properties for the leader lines, it is often easier to at first to prototype with the "Simple Line Callout" type of leader.  The leader tolerance property is the number of distance units that must be exceeded in order to permit the leader line to be drawn.  Since very short leader lines are distracting, I usually set this to something like 12 points.  So the trick is to ensure your labels are sufficiently far offset from your featues to cause the leader lines to draw.  Your example showed line features being labeled, so that's easy--just set the offset property that is shown in the label manager to something like 15 or 20 points. 

Experiment with the settings using the Simple Line Callouts--it's easier to test and refine the look of how the labels work on the map in this way.

If you need to use the more advanced leader line options, change to the Line Callout and things get a bit more complex.  There is something called the gap property there, which is the distance from the leader line to the label.  There are also margins.  I've never liked the default settings for either of these--preferring values ranging between 0 and 2 depending on what I'm labeling.  I also tend to turn off the border and accent bar (which are the basis for why the margins and gap settings are larger than my preferences).  

So, the main thing is to make sure your offset distance exceeds your leader tolerance setting and you will see leader lines.  One other variable with this item is how the labels are offset--depending on the type of feature (point, line, polygon) try different Label Offset properties in Maplex (on the Label Position tab) and see what works best.