I tried adding in a few other Content Editor Web Parts and for the most part they broke in the same way. I closed them using the Web Parts Maintenance Page (which is like the regular page only you add ?contents=1 to the URL) and then had a look at the original web part. Clicking Source Text Editor, I noticed the following.
<DIV blah blah blah piles of attributes that looked as if they'd been lifted out of MS Word etc etc>
Since the DIV contained no text or formatting, I deleted it and refreshed the web part. Hey presto, the borkedness was fixed. The tags were interfering with the HTML. Interesting indeed if this is being caused by SharePoint's very own Rich Text Editor!