Not really sure. I just heard that suggested before, so I just wanted to contribute that. I know for Public Health it worked. Had really modest undergrad grades (<3.00), waitlisted at a top School of Public Health. Wrote one where I detailed what my goals were and how this school would help me get there. I got real specific about how this school really a fit for me. It worked for me. Did I follow through on those plans? No. The only thing that I did do from that letter was to do everything in my power to be the best student I can be; went from barely in to scholarship and published. It worked once for me to get my dumb butt into a top 5 public health program. Really turned it around and now get to choose a dental school. I think genuine passion bleeds through, and if a door is left just a little open then you should go for it.
If I were you, perhaps give it a week or 2, but if (in your heart of hearts) think Columbia is for you, then you owe it to yourself to have tried everything you possibly could to give yourself a chance.