We present precise stellar radial velocity measurements of Gamma Dra taken from 2003 to 2017. The data from 2003 to 2011 show coherent, long-lived variations with a period of 702 d. These variations are consistent with the presence of a planetary companion having m sin i = 10.7 M_Jup whose orbital properties are typical for giant planets found around evolved stars. An analysis of the Hipparcos photometry, Ca II S-index measurements, and measurements of the spectral line shapes during this time show no variations with the radial velocity of the planet which seems to confirm the presence of the planet. However, radial velocity measurements taken 2011 -- 2017 seem to refute this. From 2011 to 2013 the radial velocity variations virtually disappear only to return in 2014, but with a noticeable phase shift. The total radial velocity variations are consistent either with amplitude variations on timescales of ~ 10.6 yr, or the beating effect between two periods of 666 d and 801 d. It seems unlikely that both these signals stem from a two-planet system. A simple dynamical analysis indicates that there is only a 1-2 % chance that the two-planet is stable. Rather, we suggest that this multi-periodic behavior may represent a new form of stellar variability, possibly related to oscillatory convective modes. If such intrinsic stellar variability is common around K giant stars and is attributed to planetary companions, then the planet occurrence rate among these stars may be significantly lower than thought.
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