If normality assumption is provoked, you can still perform repeated measures ANOVA provided that the sample size is large enough. If not, you would use Friedman's test.

If normality can be assumed or not has nothing to to with the sample size. The degree of reasonability of that assumption follows from the understanding of the response variable and the processes leading to the data (the "data-generating process"). Variables that represent concentrations or intensities (gene or protein expression and alikes) are usually assumed log-normal distributed (there is vast theoretical and empirical evidence available in support). Time-to-event variables are typically assumed Gamma- or Weibull-distributed, count variables (e.g. number of clones growing on a plate etc) are assumed having a Poisson or negative-binomial distribution. Juts to give a few common cases.

Of course, it is recommended to cross-check the actually observed data are not grossly contraditive of a presumably reasonable assumption. Usually one uses residual diagnostic plots for this purpose or compares information critera of models fit that use different distributional assumptions. But this is possible only whe there is a reasonable amount of data available. Your sample size might be too small to go this way, so all you have is your understanding of the variable / data-generating process.

In any case it is NOT recommended to use separate unrelated t-tests to compare conditions, because the information about the variance is then used only from the two groups being compared, and all the valuable information that is provided by all other groups is ignored (note that t-tests are used when the variance is needed for the test and has to be estimated from the observed data). It is much better to estimate the variance from all the data, using a model that includes all the groups (a special case of a model assuming a normal distributed variable and with the predictor being a categorical variable would be known as an ANOVA model, as Yasser Al Zaim wrote; other options are for instance Gamma or "quasi" models that also need to estimate a variance parameter), and perform so-called post-hoc tests that use this "pooled variance estimate". Again, in the case of a normal distributed variable these would be post-hoc t-tests.

This procedure uses the additional assumption that the variance is independent from all predictors in the model (e.g. that the variance is identical in all your groups). Under the (vvalid) assumption of normal distribution, this should also be the case. If it is not the case, then this is a strong hint that you are missing something in your model, that the data from the different groups are not really comparable because they are obtained under systematically different conditions. In biological experiments, inhomogeneous variance ("heteroscedasticity") is typically a sign that the assumtion of a normal distribution is not reasonable. You often observe that the variance and the mean are positively correlated - a stron hint that the log-normal distribution is a much more reasonable assumption. I don't know any practical case where the normal assumption under the presence of heteroscedasticity would make sense. But if you'd encounter such a strange case, you might consider using regression weights, what is still better than using individual variance estimates.

Eventually, if you have several (2+) groups, you might be interested in screening for some difference between some of the groups. In this case, chosing the "significant" p-value amoung the set of tests performed will accumulate the type-I errors of all the individual tests (including those that failed to reach significance), and this must be considered for the interpretation of the selected p-values. Typically, this is done by a correction for multiple testing: either a more stringent level of significance the p-values are compared to is chosen, or teh p-values are multiplied by a correction factor. There are many different ways a correctionfor multiple correction can be done. The simplest is the Bonferroni-correction, and this is typically sufficient when the number of tests is relatively small (3-5). With more tests, the Bonferroni-Holm procedure has more power. And there are further, computationally more complicated procedures like for instance Tukey's HSD procedure.

thanks for your help