The conditions under which the Central Limit Theorem hold are complicated (in particular, weaker and weaker conditions have been shown to be sufficient even fairly recently). But intuitively, they boil down to two conditions that are fairly sensible:<p>1. The individual observations can&#x27;t be <i>too extreme</i>. Bounded variance, for example, is sufficient, but weaker conditions suffice.<p>2. The observations must be <i>somewhat independent</i>. Independence is a strong condition and sufficient, but again weaker conditions suffice.<p>If 1) doesn&#x27;t hold, a massive outliers could swamp all other observations. If 2) doesn&#x27;t hold, for example if one underlying factor influences all observations, all bets are off: the observations could &quot;conspire&quot; to be fairly large (or fairly small), thus giving rise to fat tails inconsistent with the normal distribution.<p>(Having IID (independent identically distributed) observations, btw, is not sufficient, if the distribution itself is too fat tailed.)