BuddyPress.org

This sometimes happens to me when one of the plugin-created widgets is coded incorrectly, which messes up the way that the drag-n-drop javascript works on some browsers. Try dragging the very first widget on your widget list – does it work? If so, try the second, and the third, etc until you find the one causing the problem. Disable the plugin that creates the widget in question.

Also, I’ve found that Firefox/Camino are less likely to have the problem than Webkit browsers (Safari, Chrome).