If you are dragging a part with the mouse when you delete it with the [del] key, you lose all ability to drag other parts, either off of your ship or off of the part list on the left.
Steps to reproduce:
1) Start a new rocket.
2) Click the engine on the bottom and drag it off of the ship. Do not let go of the mouse button.
3) Hit the [delete] key to delete the part.
4) Notice that you can no longer drag parts, though you can select them.
5) Exiting the editor and re-entering solves the problem until it occurs again.
NOTE: Screenshot is not related, as I can not report the problem while dragging a part, to get the screenshot.
This post has been merged into this post and this post has been marked as duplicate.