I will say that I've looked at a question called, "Distance constraints between objects/vertices/edges," but it didn't help.
I am not sure if this is a known bug or if it is meant to be this way, but when I add two Limit Distance constraints to an object (empty), the second negates the effect of the first. I would like it if when the empty translates to the maximum distance between the two objects, the objects stop separating.
For example, I was working on a project to answer a question concerning hinged objects and I had almost everything worked out. For the bottom left joint, connecting the triangle to the bottom bar, I used an empty with Limited Distance from the bottom bar and Limited Distance from an empty at the left of the top bar (I would Stretch To the triangle mesh between the two aforementioned empties). The following is the image the user posted in the second question linked.
This is the .blend file I was working on: