I've searched for this issue, but can't seem to find anything similar to the problem that I am having.
I've recently installed a new theme, but didn't add any widgets.
Today I went to add the widgets, and I can only add one at a time!
When I click on "Add" the widget is added, the yellow box appears saying "Don't forget to save". When the box appears the "Widgets", "Available Widgets" and the bar with the drop down box moves down the page.
The bar with the drop down box moves *under* the list of available widgets.
Some how this renders the "Add" links as unclickable.
I have found two ways to fix this.
The first is add one widget, then save.
The second way is to add a widget, then click on the "Edit" link on the widgets in the "Current Widgets" list.
This then moves the list of available widgets down and the links start working again.