Does increasing the sample size always increase the accuracy of a consistent estimator?

Birnbaum (1948) introduced the notion of peakedness about \theta of a random variable T, defined by $P(| T - \theta | <\epsilon), \epsilon > 0$. What seems to be not well-known is that, for a consistent estimator Tn of \theta, its peakedness does not necessarily converge to 1 monotonically in n. In this article some known results on how the peakedness of the sample mean behaves as a function of n are recalled. Also, new results concerning the peakedness of the median and the interquartile range are presented.