Is it possible that you ran out of power? When the power goes out, other rooms (such as the cafeteria and the water treatment plant) risk going offline, and when they power down, they don't produce resources, regardless of how many people are in them.
Radiation in the vault comes from lack of Water, and Sickness (low HP) comes from lack of Food. Thankfully, dwellers don't seem to die from lack of water or food. Unhappiness (which results in loss of productivity) can come from a variety of things, and one of those things is working next to dead dwellers. If those dwellers were in the Power room, it's possible that power output was not at capacity, and with a growing vault, that can easily lead to catastrophe.
As far as hints to recover the vault:
If you have caps to spare, make sure all of your power rooms are triple-wide. Build a power room right next to another power room (no elevators in between) and they will join, up to three rooms wide. Really, all of your production rooms should be triple-wide, if you can afford it. After they are all triple-wide, you can work on upgrading them.
After you're out of caps:
Click the bottlecap icon in the upper left. This will bring up your vault roster. Click the title of the SPECIAL column to sort your dwellers by Strength. Click the dweller with the highest strength, and assign them to the power room. Continue in this fashion until the power room is filled, and do the same for any other power rooms. Change the SPECIAL column to Agility or Perception and fill your food and water rooms with whoever isn't in a power room. Don't worry about any other rooms you might have at this point.
Next:
Collect any finished resources, and immediately close the game. Double-tap your device's home button, and swipe the app upward. This will ensure that no disasters occur while you wait for resources to be generated. Assuming the highest-level people are in each room, this should only take 2-3 minutes at most, but you don't want to take any chances when your people are at such low health. Every minute you spend in game is another minute raiders or roaches could wipe your vault.
If an incident does happen:
Immediately move your dwellers out of the target room(s). Once the incident has passed out of any room, move as many dwellers as will fit INTO that room, because the incident will not return to that room. After the incident passes, everyone will return to the room they were in before the incident happened, so don't worry about trying to reassign them afterward, just continue where you left off.
Good luck! It might seem difficult at first, but I'd once rebuilt a massive vault where all but three people died, so you can do it! Just remember: don't remove dead vault dwellers who are wearing items if you can help it, because you'll lose those items! Revive those dwellers if you can!
