Find ShowCell.doc in the attached zip. The red bordered cell is the "middle outer cell" I mentioned above and it's however a table cell, not a paragraph. The green bordered table cells contained in it should be programmatically made wider than the container. I've applied the provided code to your document and then converted it to HTML using Aspose.Words. The result is TestMissedBorder Out.html - it seems to be correct, no borders disappear. However, if you do the same utilizing MS Word, the produced document does contain the problem. This is yet another evidence that Aspose.Words has nothing to do with the issue and this is a browser bug or hitch.
Please understand us correctly, the HTML produced by Aspose.Words is completely valid. We've been trying to help you to overcome the browser rendering issue; however, it is out of our responsibility as this is not the bug of our component. So if nothing helps, you should try to find a solution yourself. The easiest way is to restructure the document I think; maybe use other ways to position the document contents instead of nested tables.