Take the SDP program 

\begin{align*}
\min~&  x_3\\
s.t.~&
X = \begin{bmatrix}
        x_1   &        x_2 &    0      &     0    \\
        x_2   &        0   &    0      &     0    \\
        0     &        0   &    x_2    &     0    \\
        0     &        0   &    0      &   x_3-2  \\
\end{bmatrix}\succeq \mathbf{0}\\
&x_1,~x_2,~x_3\in\mathbf{R}
\end{align*}
It's dual can be written as follows (see below why).
\begin{align*}
\max~&  2z_{44}\\
s.t.~& Z=
\begin{bmatrix}
        0     &        0   &    0      &     0    \\
        0     &     a^2+\Delta^2&    0      &     a    \\
        0     &        0   &    0      &     0    \\
        0     &        a   &    0      &     1  \\
\end{bmatrix}\\
&Z\succeq 0
\end{align*}

Now notice that any feasible solution $\mathbf{x}$ of the primal satisfies
$x_2=0$ because of the zero situated at position $(2,2)$; recall that any SDP matrix $A$ such that $A_{ii}=0$ has only zeros on row and column $i$.

Now check that $X$ and $Z$ share the eigenvector 
$
\left[\begin{smallmatrix}
        0     &
        0        &
        1        &
        0        \\
\end{smallmatrix} \right]^\top$ with eigenvalue 0, which actually answers the question. *No* linear combination of the rows of $X$ and $Z$ that can be equal to this eigenvector, *i.e.*, the rows of $X$ and $Z$ do not cover the whole space $\implies rank(X)+rank(Z)\leq n$.

The only detail that remains to be filled is to show that the above expression of the dual is correct. Any feasible $Z$ satisfies $z_{11}=0$ because the coefficient 
of $x_1$ is zero in the primal objective function. This forces row 1 and column 1 of $Z$ to have only zeros. The dual
constraint corresponding to $x_2$ is $2z_{12}+z_{33}=0$; since $z_{12}=0$, we
have $z_{13}=0$. The dual constraint corresponding to $x_3$ imposes $z_4=1$.
There is no constraint on $z_{24}=a$; $z_{22}$ needs to be greater than or equal
to $a^2$ so as to have a non-negative principal minor corresponding to
rows/columns 2 and 4; we can write
$z_{22}=a^2+\Delta^2$. Finally, both programs have objective value 2.