Blender cannot snap to mid points unfortunately, only the knife tool exclusive snapping system can snap to midpoints as far as I know.
As of Blender 2.81 mid point snapping for mesh edges is available in Blender from the Snap Popover in the 3D View header.
For previous versions, a possible workaround is to use Blenders Mark Multiple Snap Targets feature to simulate a midpoint.
You will have to make sure snapping is forcefully turned on from the active magnet icon, and also set to Vertex and Closest
Then while transforming you will no longer need to press Ctrl to snap to other vertex. That leaves you free to press A while inside transform modal operators to "acquire targets" for snapping.
If you hover two different vertex at the end of the same edge and press A while each vertex is highlighted, one at a time, it will effectively snap to the midpoint of those two vertex.
Before you confirm transform make sure you are hovering no other vertex at all (including previous) or it will average the snap including those as well.
More about it in Blender Manual