Short answer: no. Long answer: maybe. For a distribution with finite mean and finite, non-zero variance, the sample mean is asymptotically normally distributed (by the central limit theorem). In practice, for "reasonable" variances, the convergence is fast enough that just assuming the sample mean is normal is often OK. But it is not a given. The finite variance is important. There are distributions with non-finite variances and one needs to be careful dealing with them.
