How can I save my office plant from dying?

Pictures:

Brown tips.

General die-off.

Background
We got this great bushy plant for the Fluther office from Home Depot. Unlike the other plant, it is clearly suffering. I have no idea what’s wrong with it, or how to save it.

It’s right next to the window that gets plenty of light. I haven’t been watering it very much, because I’ve previously killed plants by over-watering. The soil never felt too dry, but honestly I have no clue how much it needs.

It was fine for a few weeks, then the tips all got brown/black, which clearly seemed bad, though I don’t know what it means. Coming in today, whole branches are dying off.

What is wrong with it, and what can I do to save it?