  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 $\sim$ 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.
