Case studies are one of many examples of valid scientific methods that do not use statistics. While we canwe can draw limited statistical conclusions from samples of size one, science not always uses statistics.
As far as I know, such methods are used for example in neurology where case reports of rare brain injuries has helped us to learn about human brain. In many such cases experimental studies would not be possible for technical and ethical reasons and large-sample observational studies are impossible since prevalence of such injuries is too small.
Notice that on grounds of logic it is often enough to have single case to disprove something (e.g. you need single flying machine to disprove that "heavier-than-air flying machines are impossible").
Moreover, in some disciplines of science (I have in mind mostly social sciences) observational studies are the only possible, imagine for example cultural anthropologist observing some small tribe in the middle of amazonian forest. In such cases you can make limited use of statistics, but you would mainly depend on non-statistical methods.
