The idea of k-anonymity is that every database record appears at least k times. If you have a lot of records and few fields, your value of k could be high. But as you get more fields, it becomes more likely that a combination of fields is unique. If k = 1, then k-anonymity offers no anonymity.
Another problem with k-anonymity is that it doesn’t offer group privacy. A database could be k-anonymous but reveal information about a group if that group is homogeneous with respect to some field. That is, the method is subject to a homogeneity attack.
This is intended to be a “get you thinking” type of post, and John does have links to related posts which flesh things out a bit more.