Here is a recent result from FUN 2012 paper Picture-Hanging Puzzles by Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest and Mihai Patrascu.

We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed.

Don't let the 'polynomial number' fool you...it turns out to be $O(n^{43737})$.