Reasons for data to be normally distributed What are some theorems which might explain (i.e., generatively) why real-world data might be expected to be normally distributed?
There are two that I know of:


*

*The Central Limit Theorem (of course), which tells us that the sum of several independent random variables with mean and variance (even when they are not identically distributed) tends towards being normally distributed

*Let X and Y be independent continuous RV's with differentiable densities such that their joint density only depends on $x^2$ + $y^2$. Then X and Y are normal.
(cross-post from mathexchange) 
Edit:
To clarify, I am not making any claims about how much real world data is normally distributed. I am just asking about theorems that can give insight into what sort of processes might lead to normally distributed data.
 A: In physics it is CLT which is usually cited as a reason for having normally distributed errors in many measurements. 
The two most common errors distributions in experimental physics are normal and Poisson. The latter is usually encountered in count measurements, such as radioactive decay.
Another interesting feature of these two distributions is that a sum of random variables from Gaussian and Poisson belongs to Gaussian and Poisson. 
There are several books on statistics in experimental sciences such as this one:Gerhard Bohm, Günter Zech, Introduction to Statistics and Data Analysis for Physicists, ISBN 978-3-935702-41-6
A: Many limiting distributions of discrete RVs (poisson, binomial, etc) are approximately normal. Think of plinko. In almost all instances when approximate normality holds, normality kicks in only for large samples.
Most real-world data are NOT normally distributed. A paper by Micceri (1989) called "The unicorn, the normal curve, and other improbable creatures" examined 440 large-scale achievement and psychometric measures. He found a lot of variability in distributions w.r.t. their moments and not much evidence for (even approximate) normality.
In a 1977 paper by Steven Stigler called "Do Robust Estimators Work with Real Data" he used 24 data sets collected from famous 18th century attempts to measure the distance from the earth to the sun and 19th century attempts to measure the speed of light. He reported sample skewness and kurtosis in Table 3. The data are heavy-tailed.
In statistics, we assume normality oftentimes because it makes maximum likelihood (or some other method) convenient. What the two papers cited above show, however, is that the assumption is often tenuous. This is why robustness studies are useful.
A: There is also an information theoretic justification for use of the normal distribution. Given mean and variance, the normal distribution has maximum entropy among all real-valued probability distributions. There are plenty of sources discussing this property. A brief one can be found here. A more general discussion of the motivation for using Gaussian distribution involving most of the arguments mentioned so far can be found in this article from Signal Processing magazine.
A: The CLT is extremely useful when making inferences about things like the population mean because we get there by computing some sort of linear combination of a bunch of individual measurements.  However, when we try to make inferences about individual observations, especially future ones (eg, prediction intervals), deviations from normality are much more important if we are interested in the tails of the distribution.  For example, if we have 50 observations, we're making a very big extrapolation (and leap of faith) when we say something about the probability of a future observation being at least 3 standard deviations from the mean.
