Several wave scattering phenomena in optics are modeled by the method of fictitious sources (MFS). Despite its interesting features, the effectiveness of the MFS and its applicability are restricted by open issues, including the placement of the fictitious sources (FS) and the fields' convergence. Concerning these issues, we investigate here the MFS convergence and study oscillations in its solutions for a representative scattering problem of a dielectric cylinder illuminated by a current filament. It is shown analytically that, when the FS radii lie in the interior and exterior of two disks with certain critical radii, the MFS currents' series diverge while the respective fields converge, as the FS number N tends to infinity. Asymptotic formulas of the divergent currents are established, exhibiting that they increase exponentially with N and oscillate. Numerical simulations are included, demonstrating that (i) the divergent currents oscillate for sufficiently large N, (ii) the oscillating values are fairly approximated by the derived asymptotic expressions, and (iii) these oscillations are inherent in MFS and are not due to ill-conditioning; hence, they cannot be overcome by improving the hardware or software. The possibility of obtaining convergent and correct fields from divergent intermediary currents may lead to a potential significant advance of the applicability of the MFS.