punching down very thin cable

Has anyone done it and if so do you have any advice? I've only terminated the standard cable, with the solid-core copper. The really thin stuff looks pretty hard. Also, I need to terminate with RJ45 connectors. To add to the frustration it looks like Monoprice stopped selling the ends that accept stranded cable.

EDIT: CAT5 / CAT6 cable, I forgot to put that in the title. Not just random "thin" cable.