Confirmation of normality using residuals from an linear regression 95% of my residuals from an linear regression lie within 2 standard deviations of the predicted values. Is this enough to confirm normality or could any other distributions have 95% of residuals that lie within 2 standard deviations? Obviously I could plot the residuals to check normality, but I'm interested in knowing if the 95% within 2 standard deviations is enough to confirm normality.
 A: A large variety of distributions will tend to have something reasonably close to 95% of the residuals within 2 standard deviations.
For example here's the result of simulating 1000 regressions (each at n=100), with exponentially distributed errors (shifted exponential with mean 0), and computing the proportion of residuals within $0\pm 2s$:

In about 3/4 of the simulations, the proportion of the residuals within 2sd's was between 94 and 96%, ande in more than 9/10 simulations, between 93 and 97% of the residuals were within 2sd's.
So having very close to 95% of the residuals within 2sd's is pretty consistent with very non-normal residuals as well. 
An interesting fact -- for symmetric, continuous*, unimodal distributions (whose first two moments are finite) the proportion of the distribution within two standard deviations of the mean is limited to be between 88.9% and 100%... so for those cases, it's actually hard to get very far away from 95%. Lots of common distributions come out within a few percent of 95% of the distribution within 2sds, so something like my example would occur with many other choices of distribution.
* continuous everywhere, except perhaps at the mode
[The deleted mention of symmetry relates to an older inequality; there's no need for it now]
A: No, there are several reasons why 95% of residuals lying within 2 standard deviations need not imply normality.
I’ll consider cases in which the assumptions (other than normality) of the Classical Linear Regression Model hold, in particular that there is no heteroscedasticity or autocorrelation. If these assumptions do not hold there would be further complications.
Normality, in a regression context, generally refers to normality of the distribution of disturbances or error terms in a population regression model (throughout this answer I use the term ‘disturbances’ in this population-related sense).  In estimating a regression model, we usually take a sample of observations from the population of interest, and calculate the regression on the sample data.  The fact that we rely on a sample introduces considerations that can impact on any inference from the distribution of the residuals to the distribution of the disturbances.
Firstly, the sample might be selected by a biased process.  If so, then (even if the sample is large) the distribution of residuals may not give any useful information about the distribution of the disturbances.
Secondly, the sample (even if selected in an unbiased manner) will be subject to sampling error, which can be an important consideration if the sample is small.  Sampling error in this context can have two consequences.  One is that the distribution of residuals may not be representative of the distribution of the disturbances.  The sample might for example happen to include a disproportionately large or small number of outliers.  A second consequence is that the standard error of the regression $s$ may not equal the standard deviation of the disturbances $σ_0$.  This is explained further below.
The standard error of the regression is calculated as:
$$s = \sqrt{\frac{SSR}{N-K}}$$
where $SSR$ is the sum of squared residuals, $N$ is the number of observations (sample size), and $K$ is the number of parameters estimated.  It can be shown that $s$ is an unbiased estimator of $\sigma_0$ (1).  However, for an estimator to be unbiased is a repeated sample property.  It doesn’t mean that $s$ as estimated from one sample will equal  $\sigma_0$.  Considered over repeated samples, $s$ is a random variable with an approximately chi-square distribution given by (2):
$$s^2 \sim \frac{ \sigma_0^2\chi_{N-K}^2}{N-K}$$
This formula is exact when the disturbances are normally distributed.  How good an approximation it is when their distribution is not normal appears rather intractable, but I justify its use here on the grounds that I am only identifying what can happen as a result of sampling error, not what must happen.  Let’s see what the formula implies with, say, $N = 100$ and $K = 2$.  Then:
$$s^2 \sim \frac{ \sigma_0^2\chi_{98}^2}{98}$$
The upper 5% (1-tail) limit of a $\chi_{98}^2$ distribution is about 122, from which we can infer (dividing 122 by 98 and taking the square root) that the upper 5% (1-tail) limit of the distribution of $s$ may be about $1.1\sigma_0$.  So the probability that the standard error of the regression is more than 10% higher than the standard deviation of the disturbances may be about 5%.  And if the sample size is less than 100, that probability is likely to be higher.
Thus if we compare the distribution of the residuals with an interval of two standard deviations either side of zero, taking the standard deviation to be the standard error of the regression as calculated from the sample data, neither side of the comparison is necessarily a true reflection of the distribution of the disturbances.   The residuals may not be representative, and the standard deviation may not be correct.  It is fair to add that these two effects will to a considerable degree be related.  Nevertheless, an inference built upon such insecure foundations (unless the sample size is large) cannot be reliable, even if it leads to a correct conclusion on some occasions.  
Finally, there is the case in which the sample is large, the distribution of residuals is representative of that of the disturbances, and the estimated standard deviation does equal that of the disturbances.  Even if all those conditions are satisfied, there is still the major problem, well explained in the answer by Glen_b, that there are many non-normal distributions meeting the condition that 95% of values lie within 2 standard deviations of the mean.
Sources:


*

*Ruud P A (2000) An Introduction to Classical Econometric Theory p
158.

*Ruud (as above) p 199.
