Tapered strokes not the right shape

I am working on an illustration that uses a lot of tapered strokes. I'm using the stroke width tool to make one end of a line wide and taper it to a point.

It works great, up until the lines are more than about 2 inches long. Then they get pointed on both ends and wide in the middle, even though they have the same width profile applied.

Has anyone else experienced this? Is there a solution, or a better way to achieve the same look?