Jodi Nearns writes,

I recently employed hierarchical generalized linear modeling for my dissertation and I did have a case in which I added a predictor to a model and the Tau of my random intercept increased. I was searching on the web and saw that you will be having a book published in the Fall that included this very topic. I was curious if you would be willing to tell me why this happens or if you could direct me to a paper that discusses this?

In classical regression, adding a predictor can only decrease the residual variance (except for the minor increase that can occur from reducing the degrees of freedom by 1 and thus dividing by n-k-1 instead of n-k in the variance calculation). But in a multilevel model, adding an individual predictor can sometimes make the group-level variance go up. This can happen when the individual-level predictor is negatively correlated with the group coefficient.

Here‘s the relevant section from the book (including a pretty picture).