Think about all the false and sometimes dangerous conclusions that come from simply multiplying probabilities, thinking events are independent. Because of all the built-in redundant safeguards, we put into our nuclear power plants experts using the independence assumption told us that the chance of a major nuclear accident was infinitesimal. But as we saw at Three Mile Island, humans make correlated errors especially when they are in a panic because of one initial error which quickly can compound itself. It might be difficult to construct a realistic multivariate model that characterizes human behavior, but realizing the effect of a horrible model (independent errors) is clear.
There are many other examples possible. I will take the Challenger Shuttle disaster as another possible example. The question was whether or not to launch under low-temperature conditions. There was some data to suggest that the o-rings could fail at low temperatures. But there was not a lot of data from past missions to make it clear how high the risk was. NASA has always been concerned with the safety of the astronauts and many redundancies were engineered into the spacecraft and launch vehicles to make the missions safe.
Yet prior to 1986, there were some system failures and near failures probably due to not identifying all possible failure modes (a difficult task). Reliability modeling is a difficult business. But that is another story. In the case of the shuttle, the manufacturer of the o-rings (Morton Thiokol) had done some testing of the o-rings that indicated the possibility of failure at low temperatures.
But the data on a limited number of missions did show some relationship between temperature and failure but because redundancy led some administrators to think multiple o-ring failures would not happen, they put pressure on NASA to launch.
Of course, there were many other factors that led to the decision. Remember how President Reagan was so anxious to put a teacher in space so as to demonstrate that it was now safe enough that ordinary people who were not astronauts could safely travel on the shuttle? So political pressure was another big factor affecting the decision. In this case with enough data and a multivariate model, the risk could have been better demonstrated. NASA use to try to err on the side of caution. In this case, putting off the launch for a few days until the weather warmed up in Florida would have been prudent.
Post-disaster commissions, engineers, scientists and statisticians did a great deal of analysis and papers were published. Their views may differ from mine. Edward Tufte showed in one of his series of books on graphics that good graphics might have been more convincing. But in the end, although these analyses all have merit I think the politics would have still won out.
The moral of these stories is not that these disasters motivated the use of multivariate methods but rather that poor analyses that ignored dependence sometimes lead to gross underestimates of risk. This can lead to overconfidence which can be dangerous. As jwimberley pointed out in the first comment to this thread "Separate univariate models ignore correlations."