Found the answer.  So the issue was that even though the interface functions are pretty similar to excel, cell borders are handled like individual objects with properties.

Meaning two adjacent cells don't share a common border definition between them at the boundary.

So in order to remove the bottom border on the top cell, you have to remove the bottom border on the top cell *AND* the top border on the middle cell.

![Removed Borders](http://imgur.com/TH6Y3nC.png)

What I really want to do is merge column headers vertically, but that doesn't appear possible in SSRS.  One work around presented in the quest [SSRS tablix Merging cells vertically problem][1] was to just remove the header borders.

However, when I remove the bottom border (or change the border at all for that matter), the old one still remains.

For example, here's a test case where I've removed the bottom border and add a wide red border to the top and sides:

![Tablix](http://i.imgur.com/L5suhYg.png)

And here are the properties on the top text box itself:

![Properties](http://i.imgur.com/f2OnRP9.png)

*How do I get rid of the black lines?*


[1]: https://stackoverflow.com/a/23300962/1366033