Radial-velocity variations of the K giant star Aldebaran (\(\alpha\) Tau) were first reported in the early 1990s. After subsequent analyses, the radial-velocity variability with a period of \(\sim 629\,\mathrm{d}\) has recently been interpreted as caused by a planet of several Jovian masses. We want to further investigate the hypothesis of an extrasolar planet around Aldebaran. We combine 165 new radial-velocity measurements from Lick Observatory with seven already published data sets comprising 373 radial-velocity measurements. We perform statistical analyses and investigate whether a Keplerian model properly fits the radial velocities. We also perform a dynamical stability analysis for a possible two-planet solution. As best Keplerian fit to the combined radial-velocity data we obtain an orbit for the hypothetical planet with a smaller period (\(P=607\,\mathrm{d}\)) and a larger eccentricity (\(e=0.33 \pm 0.04\)) than the previously proposed one. However, the residual scatter around that fit is still large, with a standard deviation of \(117\,\mathrm{ms}^{-1}\). In 2006/2007, the statistical power of the \(\sim 620\,\mathrm{d}\) period showed a temporary but significant decrease. Plotting the growth of power in reverse chronological order reveals that a period around \(620\,\mathrm{d}\) is clearly present in the newest data but not in the data taken before \(\sim\) 2006. Furthermore, an apparent phase shift between radial-velocity data and orbital solution is observable at certain times. A two-planet Keplerian fit matches the data considerably better than a single-planet solution, but poses severe dynamical stability issues. The radial-velocity data from Lick Observatory do not further support but in fact weaken the hypothesis of a substellar companion around Aldebaran. Oscillatory convective modes might be a plausible alternative explanation of the observed radial-velocity variations.