Fiber optics works because the light moving along the fiber is kept inside it - it is reflected by the shininess of the surface of the fiber. This is "total internal reflection" - or at least for standard acrylic fiber, it's a high percentage of internal reflection, and that's good enough.
Therefore, to get light out of the fiber at any point other than the end, you will have to disturb that internal reflection. A very easy way to do this is to simply crimp the fiber beyond a certain angle, and as you can see, the remainder of the fiber will light up much more brightly.
You have to experiment with each thickness of fiber and each brightness of LED light source, but in general with a good high-bright LED you should be able to bend the fiber about 6 times without losing too much light out of the end of it.
I am using 0.75mm plain acrylic hobby fiber, and a strand of flashing color changing RGB high-bright LEDs.
This is the foundational concept that will allow you to make your own LED fiber projects - especially wearables. I've made cloaks and tutus that light up using this idea. Have fun!
PS if you need to figure out how to get your fiber attached to your LED, look here: http://youtu.be/zTD4xzlwiRU