Change the font size/zoom settings of the browser to a step or two smaller than the default. Open the file picker. Click the "Add File" button. If you have not yet used the file picker on this site, you may need to choose one of the repositories. The window does not display as expected. A screen shot can be seen at https://moodle.org/mod/forum/discuss.php?d=251580.

I've reproduced this in Firefox 26, IE 10, and Chrome 32.