Yes.

Unless the distribution is extremely strange/skewed. In such extreme cases a sample size of 200 might still be too small to get a sufficiently good approximate normal distribution of the estimate.

It is worth including the citation for the quotation so readers might understand the context. Several authors, from Tukey to Wilcox (Wilcox's recent textbooks and review papers will be the most accessible), have points out that having large outliers and heavy tailed distributions affects the power for applying tests assuming the normal distribution. 

What Ahmed posted is an excerpt from the following web-page:

Statistics Solutions. (2013). Normality [WWW Document]. Retrieved from http://www.statisticssolutions.com/academic-solutions/resources/directory-of-statistical-analyses/normality/

https://www.statisticssolutions.com/normality/

In many applications data violate normality assumptions due to asymmetry (skeweness) in the underlying distribution. Consider any environmental chemistry data where right tailed distributions prevail and the levels of the predictors are not fully controlled by the experimenter.   Often both X and Y are right tailed, resulting in a large cloud of data at the low end and a very small number of observations at the high end. These high end values have a great degree of leverage and cause havoc with the estimates, even in the presence of high R-squared and very small p-values.  So I would argue that it is not all about p-values.

In this case a very "wrong" model may look very good statistically

@John:

'In this case a very "wrong" model may look very good statistically'

What do you mean with this?

The diagnostic residual plots for a standard linear model on your data will surely look desastrous - and this is what I would say is "looking very bad statistically".

However, I suspect you meant that a p-value obtained from the test of some null-hypothesis within the model may be tiny ("highly significant") and that this would mean that the "model is looking good statistically". But that has actually the opposite meaning (more or less): a small p value tells us that the observed data is unlikely under the given model (restricted at the null hypothesis). This is neccesarily the case when the (unrestricted) model is already not adequate to describe the data. We can only interpret the p-value w.r.t. the null hypothesis when we assume that the (unrestricted) model is in fact a good model to describe the data.

And even if the unrestricted model is adequate, a low p-value means that the resricted model is not adequate to describe the data (and, therefore, the restriction should be "rejected" and the full model should be used) - so my association is: "low p-value -> the (restricted) model is bad statistically", what is usually a good thing scientifically!

I seem picky about words here. Maybe. But I think the confusion of "low p-value == good model statistically" is harmful for the understanding. Maybe you even didn't meant this anyway, but other readers could think so...

 I was referring to this aspect of the question:

" It is only important for the calculation of p values for significance testing"

I was assuming he is talking about testing the significance of a regression coefficient, such as H_0: B1=0 in simple regression.  My point is that non-normal distributions, particularly when the predictors are not set in an experimental setting with replication can also cause coefficient estimates to be biased, despite low p-values and high R-squared.. terms many incorrectly use to infer a good model.

According to Judge et al. loss of the normality assumption only affects least squares' finite sample property of efficiency. The other properties such as unbiasedness, consistency and asymptotic efficiency remain. If normality is violated, one may transform the data or use other methods such as GMM that are forgiving. It is true not to worry about loss of the normality assumption in large samples (following the CLT), but it depends on how large is large. Well, the person was correct (conventionally, N=200 is large...) However, formal tests of  normality would still be needed to be conducted and any transformations of the data would be required if the assumption did not hold. 

you have to check normality, even your sample size is million. I studied triglycerides data in my MSc thesis and the data was still highly skewed with 59,274 samples. Skewness is a character of the data. It shall not be a problem you can overlook in any case.

Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...

@Guven, the point is not that skewness will vanish in the data, but that it will become necligible in the distribution of a sufficient test statistic calculated from the data.

@Jochen, for continuous data, the point is always tends to be skewness. It is almost always; overlooked or 'reconstructed' by transformation in order to make it "handsome" as it is normal distributed. NO! The normal distribution RARELY exists is nature. This fact cannot be overlooked or reconstructed...

The normal distribution is a model. It is not the point whether or not it exists in nature. It does not exists (not only rarely - it never exists!), because it is a model. Like a "circle" does not exist - it is a just model.

The key question is if a model is appropriate to describe a particular aspect of reality.

@Jochen, good response

@Jochen, Thank you for your explaination. That is what I tried to say in a different way. Having a sample size greater than 200 RARELY provide that your data can be described by the model (normal distribution). 

As @Jochen points out, there is confusion here about how normality assumption works in general.  Step back to simple Student's T Interval. 

To generate an interval for the population mean we need to understand the sampling distribution of the sample mean. In regression we need to understand the sampling distribution of the regression coefficients- which are one in the same when the regressor is a two group categorical variable--so it is the same problem in general.

For small sample size, when underlying population is normally distributed, the sample mean has a known Student's T distribution. We build a T-Interval.

For small sample size without normal distruibution all bets are off and we use things like bootstrap intervals, or Tchebysheff's theorem for nonparametric intervals, but generally no guarantee of attaining nominal coverage!

If sample sizes are "large enough" irrespective of the data distribution, the central limit theorem implies normality of the sample mean and we create a Z-Interval, which is equivalent to a T interval for large N.  

We only need normal distribution of underlying population when sample size is small.

Maybe one more comment :)

The t-test is based on the assumptions that the data are i.i.d. normal, so that the distribution of the sample means are also normal and the distribution of the sample variance is chi-squared. These are assumptions from which the math of the test is derived.

If the data was in fact normal distributed, then the test was an exact test. However, there is nothing like normal distributed data in reality. There is only data that has a distribution that is a more or less good approximation to normal. In this (realistic) case the t-test is an approximate test. The question is only if this approximation is sufficiently good for the purpose.

Jochen, in your "one more comment" post you say almost verbatim what I tell my students.  I say to them that if we were able to sample from two perfectly normal populations with exactly equal variances, with each score being independent of all other scores, then the independent groups t-test (with df = n1+n2-2) would be an exact test.  But given that those conditions never hold, in reality it is an approximate test.  And therefore, the question we ought to be asking is, "Under what conditions will we obtain what George Box might have called a useful approximation?" 

HTH.