For the purpose of this article, 'Starvation Mode' is defined as a concept where your metabolic rate declines during the process of caloric restriction or weight loss to such a degree that further weight loss becomes impossible or weight gain occurs.
Starvation mode, according to the above definition, is for all practical intents and purposes a myth. It seems to have originated from the Minnesota Starvation Experiment, where participants were given 50% of their maintenance calories for months. They lost weight until their body simply had no source for ample calories (fat storage or food intake).
The notion that weight loss can be outright halted secondary to caloric restriction is currently unsupported, although a reduction of metabolic rate may occur. The process of losing weight itself can reduce caloric expenditure as all tissue requires some energy to exist, and most of the reduction in energy expenditure from weight loss can be accounted for from this.
In regards to case studies, there is one recorded report of an obese man fasting for 382 days (medically supervised) which was associated with a lowered metabolic rate but not a cessation of weight loss due to it.