thanks for your post, and sorry for the trouble. That's indeed weird, and I have no clue why this is happening.
Are you maybe using any user management plugins, like "Members", "Advanced Access Manager", or "User Role Editor"? Then it might be possible that these influence the user capabilities.
Another idea could be an interference of another plugin. Can you try again after temporarily deactivating all other plugins and switching to the WordPress default theme Twenty Twelve?
If that also does not help, I'd like to take a direct look at this in the WordPress admin of your site. In that case, please send me an email (the address is in the main plugin file "tablepress.php").