Version? OS? What happens when you try?
You may be getting confused by how you need to select a path to make the second cut.
When you move over a path with the Scissors tool, you should see a "cross-hairs" icon. Click to cut the path. The Scissors tool breaks a path by creating two anchor points where you click, one on top of either other. Use the Direct Selection tool to move the top anchor point to see both ends of the path.
When you want to make a second cut with the Scissors, first choose the Direct Selection tool and click on the line segment where you want to cut so that segment is selected. Move the Scissors over the segment and see the "cross-hairs" tool, then click. You should see two anchor points on top of each other. Switch to the Direct Selection tool to move them apart. Your path should now be divided into two.
THANKS - that sounds reasonable, but a bit complex, I'll try it. -d