Recently, the first detections of highly ionized gas associated with two warm-hot intergalactic medium (WHIM) filaments have been reported. The evidence is based on X-ray absorption lines due to O VII and other ions observed by Chandra toward the bright blazar Mrk 421. We investigate the robustness of this detection by a reanalysis of the original Chandra LETGS spectra, the analysis of a large set of XMM-Newton RGS spectra of Mrk 421, and additional Chandra observations. We address the reliability of individual spectral features belonging to the absorption components, and assess the significance of the detection of these components. We also use Monte Carlo simulations of spectra. We confirm the apparent strength of several features in the Chandra spectra, but demonstrate that they are statistically not significant. This decreased significance is due to the number of redshift trials that are made and that are not taken into account in the original discovery paper. Therefore, these features must be attributed to statistical fluctuations. This is confirmed by the RGS spectra, which have a higher signal-to-noise ratio than the Chandra spectra, but do not show features at the same wavelengths. Finally, we show that the possible association with a Lyα absorption system also lacks sufficient statistical evidence. We conclude that there is insufficient observational proof for the existence of the two proposed WHIM filaments toward Mrk 421, the brightest X-ray blazar in the sky. Therefore, the highly ionized component of the WHIM still remains to be discovered.