Analyzing A/B test results which are not normally distributed, using independent t-test I have a set of results from an A/B test (one control group, one feature group) which do not fit a Normal Distribution. 
In fact the distribution resembles more closely the Landau Distribution.

I believe the independent t-test requires that the samples be at least approximately normally distributed, which discourages me using the t-test as a valid method of significance testing.

But my question is: 
At what point can one say that the t-test is not a good method of significance testing?

Or put another way, how can one qualify how reliable the p-values of a t-test are, given only the data set?
The distribution of your data doesn't need to be normal, it's the Sampling Distribution that has to be nearly normal. If your sample size is big enough, then the sampling distribution of means from Landau Distribution should to be nearly normal, due to the Central Limit Theorem. 

So it means you should be able to safely use t-test with your data.

Example

Let's consider this example: suppose we have a population with Lognormal distribution with mu=0 and sd=0.5 (it looks a bit similar to Landau)



So we sample 30 observations 5000 times from this distribution each time calculating the mean of the sample 

And this is what we get 



Looks quite normal, doesn't it? If we increase the sample size, it's even more apparent



R code

x = seq(0, 4, 0.05)
y = dlnorm(x, mean=0, sd=0.5)
plot(x, y, type='l', bty='n')


n = 30
m = 1000

set.seed(0)
samp = rep(NA, m)

for (i in 1:m) {
  samp[i] = mean(rlnorm(n, mean=0, sd=0.5))
}

hist(samp, col='orange', probability=T, breaks=25, main='sample size = 30')
x = seq(0.5, 1.5, 0.01)
lines(x, dnorm(x, mean=mean(samp), sd=sd(samp)))


n = 300
samp = rep(NA, m)

for (i in 1:m) {
  samp[i] = mean(rlnorm(n, mean=0, sd=0.5))
}

hist(samp, col='orange', probability=T, breaks=25, main='sample size = 300')
x = seq(1, 1.25, 0.005)
lines(x, dnorm(x, mean=mean(samp), sd=sd(samp)))