Here's the link that comes up when you preview: http://allsafepestprevention.com/2011/giant-bugs-from-larger-oxygen-concentrations/?preview=true&preview_id=964&preview_nonce=61b8e1af55
Take off &preview_nonce to the end and it shows up for all users. Otherwise, one user gets the "You do not have permission to preview drafts" error. This user has the same permissions as the others who do NOT get the error. I've tried clearing cache, restarting and logging in as that user and still get the problem.
Also, this is on a multisite setup and this user who gets the error on one site does not get it on another.