We have used an electron beam ion trap to measure electron-density-diagnostic line-intensity ratios for extreme ultraviolet lines from Fe xii, xiii, and xiv at wavelengths of ≈185─205 and 255─276 Å. These ratios can be used as density diagnostics for astrophysical spectra and are especially relevant to solar physics. We found that density diagnostics using the Fe xiii 196.53/202.04 and the Fe xiv 264.79/274.21 and 270.52A/274.21 line ratios are reliable using the atomic data calculated with the Flexible Atomic Code (FAC). On the other hand, we found a large discrepancy between the FAC theory and experiment for the commonly used Fe xii (186.85 + 186.88)/195.12 line ratio. These FAC theory calculations give results similar to the data tabulated in CHIANTI, which are commonly used to analyze solar observations. Our results suggest that the discrepancies seen between solar coronal density measurements using the Fe xii (186.85 + 186.88)/195.12 and Fe xiii 196.54/202.04 line ratios are likely due to issues with the atomic calculations for Fe xii.