We present high signal-to-noise Keck spectra for 10 globular clusters associated with the giant Fornax elliptical galaxy NGC 1399 and compare measured line indices with current stellar population models. Our data convincingly demonstrate, for the first time, that at least some of the clusters in a giant elliptical galaxy have supersolar abundance ratios, similar to the host galaxy. From Hβ line strengths, the majority of clusters have ages of ~11 Gyr (within 2σ), which is similar to the luminosity-weighted stellar age of NGC 1399. Two of the clusters (which also reveal enhanced abundance ratios) show significantly higher Hβ values than the others. It remains unclear whether this is due to young (~2 Gyr) ages or extremely old (>15 Gyr) ages with a warm blue horizontal branch. However, a conflict with current cosmological parameters is avoided if the young age is favored. Either alternative indicates a complicated age distribution among the metal-rich clusters and sets interesting constraints on their chemical enrichment at late epochs.