In my Stats class, I was told n>=20 is the cutoff for useful information. At work, our system is programmed to classify anything under n>=30 as misleading. (We use a lot of customer surveys.)

Which of these, if either, is correct?

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These sorts of cutoffs are more "guidelines" than actual rules. –  Alex Becker Aug 19 '13 at 18:06
Yes, often 20 is sufficient to derive significant results. More important than the data-sizes are the p-values: en.wikipedia.org/wiki/P-value. –  badroit Aug 19 '13 at 18:07

For example, let's say I want to see if some function is generating random integers (lets keep it $<2^{32}$) or is generating the same number each time. Let's say I get two outputs from that function and I get 42 twice. Now I can say with high confidence (about 1 in $2^{32}$ probability of being wrong) from a sample size of only 2 that the function is not a random integer generator. I've drawn a useful conclusion with high confidence over 2 data points.
Thus, more important than the data-sizes are the p-values, which tells you the probability of the null hypothesis (in this case, the function returns the same number each time) being true. In the case above, $p = \frac{1}{2^{32}}$. By convention, if $p < 0.05$, the result is considered significant.