Applications to areas such as biology, medicine, and image analysis require understanding the asymptotics of distributions on stratified spaces, such as tree spaces. In the surprisingly common circumstance when Frechet (intrinsic) means of distributions on stratified spaces lie on strata of low dimension, central limit theorems can exhibit non-classical "sticky" behavior: positive mass can be supported on thin subsets of the ambient space. This talk reports on investigations initiated by a Working Group at the Statistical and Applied Mathematical Sciences Institute (SAMSI) program on Analysis of Object Data, and continued jointly with Stephan Huckemann, Jonathan Mattingly, and Jim Nolen.