Abstract We use the Green Bank Telescope to search for 3He+ emission from a sample of four Galactic planetary nebulae: NGC 3242, NGC 6543, NGC 6826, and NGC 7009. During the era of primordial nucleosynthesis, the light elements 2H, 3He, 4He, and 7Li were produced in significant amounts, and these abundances have since been modified primarily by stars. Observations of 3He+ in H ii regions located throughout the Milky Way disk reveal very little variation in the 3He/H abundance ratio—the “3He Plateau”—indicating that the net effect of 3He production in stars is negligible. This is in contrast to much higher 3He/H abundance ratios reported for some planetary nebulae. This discrepancy is known as the “3He Problem.” We use radio recombination lines observed simultaneously with the 3He+ transition to make a robust assessment of the spectral sensitivity that these observations achieve. We detect spectral lines at ∼1–2 mK intensities, but at these levels, instrumental effects compromise our ability to measure accurate spectral line parameters. We do not confirm reports of previous detections of 3He+ in NGC 3242 nor do we detect 3He+ emission from any of our sources. This result calls into question all reported detections of 3He+ emission from any planetary nebula. The 3He/H abundance upper limit we derive here for NGC 3242 is inconsistent with standard stellar production of 3He and thus requires that some type of extra-mixing process operates in low-mass stars.