  Fano resonances and Rabi splittings are routinely reported in the scientific
literature. Asymmetric resonance lineshapes are usually associated with Fano
resonances, and two split peaks in the spectrum are often attributed to a Rabi
splitting. True Fano resonances and Rabi splittings are unequivocal signatures
of coherent coupling between subsystems. However, can the same spectral
lineshapes characterizing Fano resonances and Rabi splittings arise from a
purely incoherent sum of intensities? Here we answer this question through
experiments with a tunable Fabry-P\'erot cavity containing a
CsPbBr\textsubscript{3} perovskite crystal. By measuring the transmission and
photoluminescence of this system using microscope objectives with different
numerical aperture ($NA$), we find that even a modest $NA=0.4$ can artificially
generate Fano resonances and Rabi splittings. We furthermore show that this
modest $NA$ can obscure the anti-crossing of a bona fide strongly coupled
light-matter system. Through transfer matrix calculations we confirm that these
spectral artefacts are due to the incoherent sum of transmitted intensities at
different angles captured by the $NA$. Our results are relevant to the wide
nanophotonics community, characterizing dispersive optical systems with high
numerical aperture microscope objectives. We conclude with general guidelines
to avoid pitfalls in the characterization of such optical systems.
