  "Is entanglement monogamous?" asks the title of a popular article [B. Terhal,
IBM J. Res. Dev. 48, 71 (2004)], celebrating C. H. Bennett's legacy on quantum
information theory. While the answer is affirmative in the qualitative sense,
the situation is less clear if monogamy is intended as a quantitative
limitation on the distribution of bipartite entanglement in a multipartite
system, given some particular measure of entanglement. Here, we formalize what
it takes for a bipartite measure of entanglement to obey a general quantitative
monogamy relation on all quantum states. We then prove that an important class
of entanglement measures fail to be monogamous in this general sense of the
term, with monogamy violations becoming generic with increasing dimension. In
particular, we show that every additive and suitably normalized entanglement
measure cannot satisfy any nontrivial general monogamy relation while at the
same time faithfully capturing the geometric entanglement structure of the
fully antisymmetric state in arbitrary dimension. Nevertheless, monogamy of
such entanglement measures can be recovered if one allows for
dimension-dependent relations, as we show explicitly with relevant examples.
