This is an obvious problem that took a few hours from my life yesterday with a fairly simple solution, but it didn’t occur to me until much later so I thought I would post here just in case it helped someone.
On a new install I created several pages, turned on permalinks, and everything was working fine. I then renamed the pages, but this caused 403 forbidden errors to occur only for those pages on the site.
After an extended diagnostic search, rebuilding permalinks, changing permissions, adding code to the .htaccess and several other things I realised that there were already directories on the server with the same name as the pages. After I renamed the directories everything started working again.
So the rather obvious but not always apparent morale of the story is to make sure you don’t have any page names the same as existing directories if you are using permalinks.