The old rule of using z or t tests or confidence intervals if n >= 30 is a relic of the pre-computer era, and should be discarded in favor of bootstrap-based diagnostics.
The diagnostics will surprise many statisticians, who don't realize how lousy the classical inferences are. For example, 95% confidence intervals should miss 2.5% on each side, and we might expect the actual non-coverage to be within 10% of that. Using a t interval, this requires n > 5000 for a moderately-skewed (exponential) population. There are better confidence intervals and tests, bootstrap and others.
The bootstrap also offers pedagogical benefits in teaching sampling distributions and other statistical concepts, offering actual distributions that can be viewed using histograms and other familiar techniques.