For years I've bumped into this and workarounds I've used are:
1. Tap the delete key.
2. Move the cursor after it and backspace.
3. In a rather stubborn document I had to use REVEAL CODES (varies with version) and delete it there.
4. Save as RTF and edit the RTF with a text editor.
Each method required more and more expertise.
I send out a weekly newsletter to parents of my students. One main and two or three small text boxes, with background colors.
Naturally, I start a new one by starting up last week's and revising it. Last week I had a big red "No school on Monday" box, and this week I want to delete it. But it won't.
When I right click I don't even see "delete" as a menu choice, and when I click on "cut" absolutely nothing happens.
This is on my Dell laptop running Word 2007 on Win 7. When I try the same operation on my desktop, the red text box goes away just fine when I "cut" it. So what's happening on my laptop?