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 that are functionally paragraph separators, but they don't cause a line break; like this:
The quick brown fox│̲̅¶̲̅│jumps over the lazy dog.¶
They allow a paragraph —
or rather, what appears to be a single paragraph —
to have multiple paragraph styles.
This is analogous to the way a “section break (continuous)”
is a section separator that doesn't cause a page break,
thus allowing a page to have multiple section formats.
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 (or the “Commands Not in the Ribbon” 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.
Or, in other words (or not?), at least in Word 2013 and 2016,
“insert style separator” is a misnomer.
This function (shortcut/button)
simply changes the following paragraph marker
(at the end of the current paragraph)
from a standard paragraph mark into a style separator,
thus joining two paragraphs into what looks like a single paragraph.
The best way to work with this might be to type Enter
where you want to break the paragraph,
then click to the left of the new paragraph mark and use the shortcut.
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.
Ok, here's a nice meaty sample document (https://docs.google.com/open?id=0B4qT-cHb903mS3k3bjZoTV95Slk) that I should be able to leave out there just about forever. It's large because it includes CC pictures as sample figures. I'm also including the notes section of it that should have enough information to recreate what I did or at least get you close to solving your issue - the document just provides a "live" example so you can look at field codes, etc. but the important values are documented here.
Chapter titles are formatted as Heading 1, and chapter numbering is done by having the cursor within the first chapter name, going to the Home tab, selecting the multi-level list button/dropdown and selecting a numbering format.
To keep Word from making the Table of Contents into Chapter 1, click into the Contents heading, select the multi-level list button, and change the formatting there to None. I did this after I had already right-clicked on the Notes chapter heading and told it to restart numbering at 1. This was a bit squirrely, as it initially had the chapter numbering in the figures off by one. Inserting a Next Page Section Break (Page Layout tab, Page Setup section, Breaks dropdown) after the Table of Contents then redoing the chapter numbering (e.g. resetting start at 1, etc.) seemed to take care of this after I changed the Heading 1 Style to have a page break before it, which caused the chapter numbering to reset to default including the Table of Contents.
Pictures were simply copied and pasted, then the text wrapping was set to Square to make them float, then they were dragged around a bit. They’re not anchored to spots in the text though they probably should be.
Picture Captions were added by right-clicking the picture and selecting Insert Caption; the caption numbering was set to Automatic and told to include chapter numbers (which is where setting chapter numbering comes in up above).
A bunch of this is actually controlled by Fields, press Alt-F9 to see the field codes. It’s likely worth noting that the field codes for my captions are “Figure {STYLEREF 1 \s}-{SEQ Figure * ARABIC \s 1}", the \s doesn’t appear to be well documented, at least not within the STYLEREF info. Additional information here: http://wordfaqs.mvps.org/styleref.htm
UPDATE: It looks like this MSDN page may be the best reference relevant to this: http://msdn.microsoft.com/en-us/library/aa163918%28v=office.10%29.aspx
In particular, the "\s 1" tells the field code to restart the figure numbering at the most recent heading of the specified level (here, level 1).
Best Answer
Your error is because the chapter numbers were not correctly assigned to begin with. You have to assign them correctly before you can begin captioning figures.
Assign chapter numbers using a 'multilevel list' along with a style. Word needs these to be correctly assigned to properly caption the figures. Go to Home > Multilevel List and assign a number to the chapter, and update its style to 'Heading 1'
When you right-click your image to caption it, click on 'Numbering' and check the option 'Include chapter number'. From the drop-down box make sure that the style 'Heading 1' is selected as shown below:
This gives a perfect caption for each image, as the following shows:
Alternatively, if you want to get better looking headings you can define your own format by going to Home > Multilevel list > Define new multilevel list. Define the number format and then customize the style (fonts, font sizes, paragraph and so on) for Heading 1.