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We consider the Pauli operator $\mathbf{H}\left(\mu \right):={\left({\sum }_{j=1}^m{\sigma }_j\left(-i\frac{\partial }{\partial x_j}-\mu A_j\right)\right)}^2+V$ selfadjoint in $L^2({ℝ}^m;{ℂ}^2)$, $m=2,3$. Here ${\sigma }_j$, $j=1,...,m$, are the Pauli matrices, $A:=(A_1,...,A_m)$ is the magnetic potential, $\mu \>0$ is the coupling constant, and $V$ is the electric potential which decays at infinity. We suppose that the magnetic field generated by $A$ satisfies some regularity conditions; in particular, its norm is lower-bounded by a positive constant, and, in the case $m=3$, its direction is constant. We investigate the asymptotic behaviour as $\mu \to \infty$ of the number of the eigenvalues of $\mathbf{H}\left(\mu \right)$ smaller than...