SD - I tried creating a new shortcut and if you delete the target you do get a more meaningful message.
While doing this, though, I noticed that the ones I'm having a problem with are slightly different - the icon is the same, but it seems like it's dimmed out, just like a menu item which is greyed out and therefore not selectable. These are all under Users and my account name, and are things like Local Settings, Recent, Start Menu, so are obviously system related.
Looking at the security settings, there is one for Everyone that allows list only, but there is also one for administrators and for my account that gives full control - so I don't why I can't even open the shortcut.
As is often the case, there must be something going on that I don't understand..........