Regarding overlapping of borders of a grouped element, @mdo commented in #24453 that it is not good way to control the border color. As an alternative, we can use border-width: 0 or negative margin.

In input group, .input-group-addon uses border width to 0, and input-group-btn uses negative margin. I think we should use a same way (negative margin), since .input-group-addon and input-group-btn are similar in principle.