It turns out the solution to this problem in Word 2003 and later is to use something called
style separators. Style separators are hidden characters like paragraph marks, but they don't break a new paragraph. Instead, they allow a paragraph to have multiple paragraph styles added to them.
Firstly, make sure you have enabled viewing hidden characters.
To insert a style separator in Word 2007 use the keystrokes Ctrl+Alt+Enter, or add the
Style separator command to the Quick Access Toolbar from the
All Commands set. When you insert a style separator, it will be added at the end of the paragraph in which the cursor is placed. Move it to the point at which you would like your caption to be split.
Don't forget to select all (Ctrl+a) and then update field codes F9 to make sure your changes get propagated into the table of figures.