  We reconsider the mean-field Potts model with $q$ interacting and $r$
non-interacting (invisible) states. The model was recently introduced to
explain discrepancies between theoretical predictions and experimental
observations of phase transitions in some systems where the $Z_q$-symmetry is
spontaneously broken. We analyse the marginal dimensions of the model, i.e.,
the value of $r$ at which the order of the phase transition changes. In the
$q=2$ case, we determine that value to be $r_c = 3.65(5)$; there is a
second-order phase transition there when $r<r_c$ and a first-order one at
$r>r_c$. We also analyse the region $1 \leq q<2$ and show that the change from
second to first order there is manifest through a new mechanism involving
{\emph{two}} marginal values of $r$. The $q=1$ limit gives bond percolation and
some intermediary values also have known physical realisations. Above the lower
value $r_{c1}$, the order parameters exhibit discontinuities at temperature
$\tilde{t}$ below a critical value $t_c$. But, provided $r>r_{c1}$ is small
enough, this discontinuity does not appear at the phase transition, which is
continuous and takes place at $t_c$. The larger value $r_{c2}$ marks the point
at which the phase transition at $t_c$ changes from second to first order.
Thus, for $r_{c1}< r < r_{c2}$, the transition at $t_c$ remains second order
while the order parameter has a discontinuity at $\tilde{t}$. As $r$ increases
further, $\tilde{t}$ increases, bringing the discontinuity closer to $t_c$.
Finally, when $r$ exceeds $r_{c2}$ $\tilde{t}$ coincides with $t_c$ and the
phase transition becomes first order. This new mechanism indicates how the
discontinuity characteristic of first order phase transitions emerges.
