Is it possible to add/delete table cells in Acrobat?

I created a table in Word so that the table cells would be recognized as fields when I bring it in to Acrobat.  So each table cell is recognized as a field, i.e. it has a field inside of it.  My last row has 3 cells in it and I want to add a 4th cell (in Word this would be like splitting a cell into 1 row and 2 columns).

In Acrobat I selected, I selected Tools>Edit PDF.  I can click and select each side of a table cell, but beyond that, I don't see how to add another table cell while in Acrobat.  Is there a way to do this?

Also, I tried grabbing the side of a cell to slide it over so that a fourth cell is created.  In the newly created space there is black in the upper half and gray in the lower half.  What is this?  What does this mean?  Is this a way to create a new cell in the table?  If so, how do I get rid of the black and gray background?