When do normal distributions not occur? I know that in many cases one can assume a normal distributed probability density. But what the situations when the distribution in non-normal. Some examples would be nice. 
For example, suppose there is a voting about when it should be legal to buy alcohol. Then one would not see a normal distribution with peak at (say) 18 years old. Rather one sees some peeks at too young and some at a little to old and so on. But if one would see a normal distribution here (say around 18 years old), what would that tell you? 
 A: Many cases of what kind? If you are taking a mean of a large sample drawn from independent identical distributions, then under certain conditions given by the Central limit theorem the distribution of the result will tend towards a normal distribution when the sample size tends to infinity. But in other cases, whatever you're talking about can have all kinds of distributions. For example, the number of failures of a machine within each time period is a discrete distribution that only takes non-negative values, and is certainly not a normal distribution. It often can be modeled by a Poisson distribution.
As for the question you added later, seeing a so-called normal distribution is a difficult problem. There are a number of Normality tests, but it still cannot really prove anything; remember that statistics can be misleading once in a while even if you are unbiased! If however you have some hypothesis about why some distribution should be normal, then these statistics can give some supporting evidence towards that claim.
A: The central limit theorem talks about instances where things are added together, roughly speaking. 
But there might be instances where you have things multiplied together instead, giving you a lognormal distribution. Gibrat's law is one famous example.
