Here is an example of the type you appear to be seeking. The story -- like many good stories -- involves a million dollar prize offered by a billion dollar corporation, sought by armies of computer wizards. There's even a courtroom scene at the climax.
In 2007, Netflix released a dataset consisting of roughly a hundred million movie ratings (from 1 through 5 stars) given by 500 thousand users. The basic idea was to award prize money to anyone who managed improve upon the existing Netflix algorithm for predicting (based on your past ratings) how much you will like a new movie. A team from AT&T research won the million dollar prize by producing an algorithm which improved the state of the art by 10% or so. Many references for those interested in the full story can be found on wikipedia here.
Of course, in order to preserve their users' privacy, Netflix "anonymized" their dataset. Meaning, they removed all mention of user names and ip addresses, etc before making the data available to the public. Much to their surprise (and their users' annoyance), a team of computer scientists from the University of Texas used publically available non-anonymous data from IMDB and deduced the identities of many Netflix users. I'm not qualified to judge the sophistication of their framework, but from a glance at their paper there certainly seems to be some non-trivial mathematics involved.
Here's an open question:
what statistical properties of your anonymized data might guarantee (with high probability) that it can not be de-anonymized?
I suspect that such considerations are likely to guide a lot of research in statistics at least in the foreseeable future