In the area of data analysis and arguably even in machine learning as a whole, few approaches have been as impactful as the classical k-means clustering. Here, we study the complexity of k-means clustering in settings where most of the data is not known or simply irrelevant. To obtain a more fine-grained understanding of the tractability of this clustering problem, we apply the parameterized complexity paradigm and obtain three new algorithms for k-means clustering of incomplete data: one for the clustering of bounded-domain (i.e., integer) data, and two incomparable algorithms that target real-valued data. Our approach is based on exploiting structural properties of a graphical encoding of the missing entries, and we show that tractability can be achieved using significantly less restrictive parameterizations than in the complementary case of few missing entries.