If you are cutting a design and you find that Origin will not allow you to change your cut type on a line from On-Line to either inside or outside cut, then the shape is probably an open path. Moving Origin across the shape, you will see the open point highlight as a green dot when it is within the corrective circle.
This often occurs when converting a design file into an svg from another file type. Try going into your design software and 'joining' any lines that you are trying to cut so that they are read as continuous. This process will differ depending on your design software of choice. In Adobe Illustrator, for instance, you will highlight the problem area and use Path>Join.