It is:
a.) possible
b.) confusing
You can control lines at the table level and at the cell level.
To change the table border, or all of the interior vertical and/or horizontal lines, click in the table and choose Table > Table Options. The border command is under Table Setup, the interior line controls are found the Row Strokes and Column Strokes tabs. The border control is intuitive, but to turn off row and column strokes you have to actually turn them on and then set them to 0. For example, to remove the row strokes, set the alternating pattern to every other row, and then set First and Next to 0.

In your case, it sounds like you need to turn off all the interior lines in Table Setup, and then add them back in via Cell Options > Stroke and Fills, which has its own peculiar interface. Cell Options uses a proxy with blue and gray lines.
So to add a stroke to say, the left side of the selected cells, click to gray out the lines that you don't want to change. In my screen shot, the top, horizontal, bottom and vertical lines are gray. The only line I left blue in the proxy is the left line. Then I can choose 5 pt Magenta and only the left side of the cell is impacted.

