This can happen if there is interfering JavaScript affecting the page. To check on this do the following:

1. Go to Site Explorer > Pages & Templates
2. Search for the page
3. Look at the parts associated. If there are Unformatted Text parts, edit them and take a look to see if there is JavaScript
4. Try cutting out any Unformatted Text parts from the page that have JavaScript in them. Do this until you can again use the content panes / edit parts from the actual page.