Data completeness is an important aspect of data quality as in many scenarios it is crucial to guarantee completeness of query answers. We develop techniques to conclude the completeness of query answers from information about the completeness of parts of a generally incomplete database. In our framework, completeness of a database can be described in two ways: by table completeness (TC) statements, which say that certain parts of a relation are complete, and by query completeness (QC) statements, which say that the set of answers of a query is complete. We identify as core problem to decide whether table completeness entails query completeness (TC-QC). We develop decision procedures and assess the complexity of TC-QC inferences depending on the languages of the TC and QC statements. We show that in important cases weakest preconditions for query completeness can be expressed in terms of table completeness statements, which means that these statements identify precisely the parts of a database that are critical for the completeness of a query. For the related problem of QC-QC entailment, we discuss its connection to query determinacy. Moreover, we show how to use the concrete state of a database to enable further completeness inferences.