I've always assumed that for most holograms, the computer intervenes to keep them from being conscious; like putting speed bumps or little walls or other forms in flowing water to stop backwash or vortices from forming.

A better analogy might be just a simple puppet. An advanced enough puppet might feed you all senses and be engaging enough to let you get carried away processing their existence as if it was your own, but if those are not your only sensory inputs, if their memories are not your only memories, if their death only sperates you from a few inputs, you would never consider yourself the puppet.

To pull this all together, what if you pulled out a few barriers, walls, etc, and vortices started forming that never access the other senses and memories; a loop is closed that cuts off the puppet from its grander existence. And now you have a new, conscious being.

Moriarty proved he was controlling the puppeteer, proving his consciousness. The doctor was cut off from the ship's puppeteer so he could process more information, and became the only animating force in his program. Sullivan, and his wife, were simply convincing puppets until proven otherwise.