ICC requires normality of data set distribution. Moreover, it shows a paradoxical inflation when is used on heterogeneous samples, with a wide range of values. Both issues are often overlooked, and the second effect is a real limitation to assess reliability. That´s the reason why it should not be employed as the sole statistic.
In my opinion, the best choice is the use of ICC and the Bland-Altman method: Whereas ICC is a global measurement of reliability, the Bland-Altman method provides an interesting information about the distribution of differences.