I have a div with border-radius set to some value (let\'s say 10px), and a nested div that is the full width and height of its parent.
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Just wanted to chime in on this one since I found this with a similar problem.
In a div with its overflow set to scroll, the border-radius didn't clip the content while scrolling unless the content was scrolled to the absolute top or bottom. Even then, the clipping only sometimes reappeared if I scrolled the document to the absolute top or bottom as well.
On a lark I added a transparent border to the element and that seemed to enforce the clipping on the corners. Since I already had some space around the element, I just cut that in half and applied the remainder to the border thickness. Not ideal, but I ended up with the result I wanted.
Tested in Chrome, Safari and Firefox on Mac.