Validity of normality assumption in the case of multiple independent data sets with small sample size Due to limitations in experimental setup, I only have small data sets with n=3. Despite the low df the difference between treated and control is large enough to generate a significant p-value.
The problem is that with small sample sizes doing a t-test becomes more sensitive to the assumption that the data are drawn from a population of a normal distribution. In my case however, multiple independent experiments consistently yield a similar result.
I cannot group the data of the experiments because of small differences in the data. For example the increase between treated and control in one experiment is slightly bigger than in another experiment, which is likely caused by small variances in experimental conditions (it would get technical to explain this further). Despite this the same increase is consistently observed and each time the 3 data points have a small standard deviation for both groups.
So my question is whether it is defensible to make the normality assumption based on the data of multiple independent experiments with a small sample size? If not am I right that it would not be appropriate to use any statistics in this case?
 A: This may help: 
DR Cox, PJ Solomon. 1986. 
Analysis of variability with large numbers of small samples. 
Biometrika 73: 543-554. 
Abstract: Procedures are discussed for the detailed analysis of distributional form, based on 
many samples of size r, where especially r= 2, 3, 4. The possibility of discriminating between 
different kinds of departure from the standard normal assumptions is discussed. Both 
graphical and more formal procedures are developed and illustrated by some data on pulse 
rates.
A: 
Due to limitations in experimental setup, I only have small data sets with n=3. Despite the low df the difference between treated and control is large enough to generate a significant p-value.
The problem is that with small sample sizes doing a t-test becomes more sensitive to the assumption that the data are drawn from a population of a normal distribution. In my case however, multiple independent experiments consistently yield a similar result.
I cannot group the data of the experiments because of small differences in the data. 

If you treat the experiments as blocks, that can be used to account for this. (Alternatively, you may want to use random effects term on intercepts, especially if this is not under control.)

So my question is whether it is defensible to make the normality assumption based on the data of multiple independent experiments with a small sample size? 

You can attempt to assess it if you assume a common error distribution and combine residuals across all experiments in order to do say a normal Q-Q plot (normal scores plot).

If not am I right that it would not be appropriate to use any statistics in this case?

You can still test your hypothesis without normality, but beware, you won't get very significant results with nonparametric tests and such tiny sample sizes. 
However, the combining experiments strategy (of using blocks) can work there too.
A: There are certainly alternative statistics. You could do permutation tests, for example. You could also do nonparametric tests, such as Wilcoxon. 
