  To certain types of generic distributions (subbundles in a tangent bundle)
one can associate canonical Cartan connections. Many of these constructions
fall into the class of parabolic geometries. The aim of this article is to show
how strong restrictions on the possibles sizes of automorphism groups of such
distributions can be deduced from the existence of canonical Cartan
connections. This needs no information on how the Cartan connections are
actually constructed and only very basic information on their properties. In
particular, we discuss the examples of generic distributions of rank two in
dimension five, rank three in dimension six, and rank four in dimension seven.
