The case study that I learned in my Numerical Analysis course was the patriot missile incident, which lead to 28 death in the early 90s. Essentially, there was a timer implemented that used floats, and due to the inherent rounding error of floats, after enough iterations this caused the missile to miss its target by miles. There were some operational errors involved, such as leaving the missile "turned on" for longer than was suggested by the manufacturer, but still, these types of incidents should not happen.

Even with double precision, there are errors. For instance, 0.3 is actually represented as:

0.299999999999999988897769753748434595763683319091796875