Minimum sample size needed to assume Normality
The below extract taken from the above link suggests that 30 observations is sufficient to make the assumption of a Normal distribution tenable.
“In general, it is said that Central Limit Theorem “kicks in” at an N of about 30. In other words, as long as the sample is based on 30 or more observations, the sampling distribution of the mean can be safely assumed to be normal”
NIHR guidelines quote Lancaster, Dodd and Williamson (2004) also suggest an overall sample size of 30 for parameter estimation such as a standard deviation.
Comparing change between two time points
The t-test, however, is at least reasonably robust to at least mild non-normality, in for example, the differences in paired t-tests (and it's the differences that are supposed to be normal no the endpoints). If the observations have small skews (say less than 1 in absolute value) and kurtoses (less than 3 in absolute value), the differences may be indistinguishable from normal even at large sample sizes. If the normality assumption holds, the t-test will still be more powerful than the signed rank test, for one. I'd imagine this is also true for various other non-parametric tests. So, it still may be best to use the t-test. That said, there's little reason to avoid the Wilcoxon (nonparametric paired) test if non-normality is the main concern.
Lancaster GA, Dodd S & Williamson PR (2004) Design and analysis of pilot studies: recommendations for good practice. J Eval Clin Practice 10 307-312.
Univariate rules of thumb for normality
In addition to the above George & Mallery (2010) suggest values for skew and kurtosis between -2 and +2 are considered acceptable in order to prove a normal univariate distribution.
George, D., & Mallery, M. (2010). SPSS for Windows Step by Step: A Simple Guide and Reference, 17.0 update (10a ed.) Boston: Pearson.