  A proof of optimal-order error estimates is given for the full discretization
of the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary
conditions in a smooth domain. The numerical method combines a linear
bulk--surface finite element discretization in space and linearly implicit
backward difference formulae of order 1 to 5 in time. Optimal-order error
estimates are proven. The error estimates are based on a consistency and
stability analysis in an abstract framework, based on energy estimates
exploiting the anti-symmetric structure of the second-order system.
