  We study the number $N_{<0}(H_s)$ of negative eigenvalues, counting
multiplicities, of the fractional Schr\"odinger operator $H_s=(-\Delta)^s-V(x)$
on $L^2(\mathbb{R}^d)$, for any $d\ge1$ and $s\ge d/2$. We prove a bound on
$N_{<0}(H_s)$ which depends on $s-d/2$ being either an integer or not, the
critical case $s=d/2$ requiring a further analysis. Our proof relies on a
splitting of the Birman-Schwinger operator associated to this spectral problem
into low- and high-energies parts, a projection of the low-energies part onto a
suitable subspace, and, in the critical case $s=d/2$, a Cwikel-type estimate in
the weak trace ideal $\mathcal{L}^{2,\infty}$ to handle the high-energies part.
