That concerns me most around places that process data for other companies (e.g., Cambridge Analytics, Facebook, Google, Amazon). These places could have access to many different data sets relating to a person, and could potentially combine these data sets to uniquely identify a single individual.
I recently looked at something that I gave a fake zip, birth date, and gender. Based on statistical probabilities it gave a 68% chance of a large data set having 1-anonymity. Wasn't clear what they were considering large, so could be bogus, but if true imagine what could easily be done with 10+ unique fields (e.g., zip, birthdate, gender, married?, # of children, ever smoked?, deductible amount, diabetes?, profession, BMI).
The earlier poster is right, only aggregate data is truly anonymous.