  Recently, there has been a surge of interest in using R\'enyi entropies as
quantifiers of correlations in many-body quantum systems. However, it is well
known that in general these entropies do not satisfy the strong subadditivity
inequality, which is a central property ensuring the positivity of correlation
measures. In fact, in many cases they do not even satisfy the weaker condition
of subadditivity. In the present paper we shed light on this subject by
providing a detailed survey of R\'enyi entropies for bosonic and fermionic
Gaussian states. We show that for bosons the R\'enyi entropies always satisfy
subadditivity, but not necessarily strong subadditivity. Conversely, for
fermions both do not hold in general. We provide the precise intervals of the
R\'enyi index $\alpha$ for which subadditivity and strong subadditivity are
valid in each case.
