This can be done only at pre-defined edit points along the curve. If you wish to split a curve at another location, you first need to insert a new edit point at that location.
To split or break a curve into two separate entities:
1. Select the Curve-Break Curve command.
2. Position the cursor over the edit point to use as the break location and click the left mouse button.
3. Repeat step 2 as many times as you want. The Break Curve command will stay active or current until you select another command.
Note: If you want to move one of the two points now at the broken ends of the curves, you need to temporarily hide the curve that you do not want to move.