To reproduce: add a widget to your dashboard, save, edit page again and add that same widget. Some widgets show some bad behavior when more than one exists on a page. The twitter widget is an example of that. The description above says the status widget is another.
The problem is that the multiple variable is not being used to prevent multiple widgets from being added to a page when a user edits a page more than once. views/default/canvas/layouts/widgets.php (or a function in the engine) would need to check what current widgets are on the page and then remove those from the widget list that have the multiple variable set to false.
Also, I wouldn't describe this as trivial for 2 reasons:
This can create some very strange behavior with widgets and there is not anything that plugin developers can do about it.
It points out there are some incomplete portions of the widget handling code. Even if the multiple flag is set in the widget handler array, it isn't used for anything as far as I can tell. If a plugin sets the multiple flag to true, a user still can't add that widget to a profile page more than once in a edit page session. (Tested with 1.2)
Related bug in views/default/canvas/layouts/widgets.php: