MS Word 2003: How to spill text from one cell into another?

Table cells don't do that directly, but text boxes do. You can have text flow from one text box to another, to another, and so on. Try putting text boxes into the table cells. You'll probably want to hide the text box borders, but wait until you have the links and sizes set up the way you want them.

Right-click on the first text box and click "Create Text Box Link", then click on the second box. Text will now flow from the first box to the second. Right-click the frame of the second box and link it to the third, and so on. When you have text flowing the way you like, then select the text boxes (click on the frame of the first, then shift-click on the frames of the other boxes to do a multiple select) and right-click a frame to get to "Format Text Box...". (Or use Format>Format Text Box...) On the "Color and Lines" tab, select "No line" for line color. to turn it off.

Hope this helps...

I am unsure what you mean by spill? Do you mean so that is works like Excel when there is empty cell to the right of a full cell? If so, you can't do that. If you mean more like an article where the text that doesn't fit at the bottom of the cell then spills into another area (not necessarily linked), then you need to use text boxes and link them.