Context. Radial-velocity variations of the K giant star Aldebaran (α Tau) were first reported in the early 1990s. After subsequent analyses, the radial-velocity variability with a period of ∼629 d has recently been interpreted as caused by a planet of several Jovian masses. Aims. We want to further investigate the hypothesis of an extrasolar planet around Aldebaran. Methods. 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. Furthermore, the possibility of oscillatory convective modes as cause for the observed radial-velocity variability is discussed. Results. As best Keplerian fit to the combined radial-velocity data we obtain an orbit for the hypothetical planet with a smaller period (P = 607 d) and a larger eccentricity (e = 0.33 ± 0.04) than the previously proposed one. However, the residual scatter around that fit is still large, with a standard deviation of 117 ms⁻¹. In 2006/2007, the statistical power of the ∼620 d period showed a temporary but significant decrease. Plotting the growth of power in reverse chronological order reveals that a period around 620 d is clearly present in the newest data but not in the data taken before ∼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. Conclusions. 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.