  Supernovae of Type IIb contain large fractions of helium and traces of
hydrogen, which can be observed in the early and late spectra. Estimates of the
hydrogen and helium mass and distribution are mainly based on early-time
spectroscopy and are uncertain since the respective lines are usually observed
in absorption. Constraining the mass and distribution of H and He is important
to gain insight into the progenitor systems of these SNe.
  We implement a NLTE treatment of hydrogen and helium in a three-dimensional
nebular code. Ionisation, recombination, (non-)thermal electron excitation and
H$\alpha$ line scattering are taken into account to compute the formation of
H$\alpha$, which is by far the strongest H line observed in the nebular spectra
of SNe IIb. Other lines of H and He are also computed but are rarely identified
in the nebular phase. Nebular models are computed for the Type IIb SNe 1993J,
2001ig, 2003bg and 2008ax as well as for SN 2007Y, which shows H$\alpha$
absorption features at early times and strong H$\alpha$ emission in its late
phase, but has been classified as a SN Ib. We suggest to classify SN 2007Y as a
SN IIb. Optical spectra exist for all SNe of our sample, and there is one IR
nebular observation of SN 2008ax, which allows an exploration of its helium
mass and distribution.
  We develop a three-dimensional model for SN 2008ax. We obtain estimates for
the total mass and kinetic energy in good agreement with the results from
light-curve modelling found in the literature. We further derive abundances of
He, C, O, Ca and $^{56}$Ni. We demonstrate that H$\alpha$ absorption is
probably responsible for the double-peaked profile of the [O {\sc i}]
$\lambda\lambda$ 6300, 6363 doublet in several SNe IIb and present a mechanism
alternative to shock interaction for generating late-time H$\alpha$ emission of
SNe IIb.
