INTERIM ANSWER:

I have been working on this problem since I started playing the game, and have come to an odd conclusion: "room types" might include having a full room (3 units) of each type.

I have tracked my Vault's progress on this spreadsheet, and have clearly indicated:

• The 19 types of rooms, and their 3 subtypes
  • e.g., the 3 levels for Diner are: Diner, Restaurant, Cafeteria.
• The number of each rooms I currently have in my vault.
• If 1 or more are present, a 1 gets placed in the "Present" column.
• This column is summed and divided by 57 (3 room types times 19 rooms), and presented as a percentage.

I honestly thought that once I had one of each of the room types, the achievement would unlock. However, this was not the case. At my current loadout, Game Center reports that I am only 40% complete for the achievement.

This has me thinking that not only do we require 1 of each room type, but they all must be of size 3 (many of my level 3 rooms are only size 1). I have now started tracking the sizes of my rooms. My next step is to get a size 3, level 3 of each room type and see how that affects the percentage.

The only thing which could explain it (besides a bug) is that you may missing some electricity or water. If you haven't enough electricity your food and water supply will fall down and maybe collapse. This will increase the radiation of your dwellers. In some cases they may die on this.

Otherwise, I would say it could be a bug. I've had a situation which was nearly the same. In my case electricity was too low, which leads to the situation I described above.