This paper studies the error that occurs when measuring surface currents with a current meter mounted on a buoy or a mooring line whose horizontal and vertical motions respond to the presence of waves. The error is defined with respect to an Eulerian reference measurement where the sensor does not move. First, we present the subject with a theoretical analysis in the case of a monochromatic wave. That idealized model allows us to study particular sensor or mooring line motions. Second, a realistic numerical model is implemented to reconstruct the current field with a high resolution near the surface. Wave orbital velocities are generated with a random phase model. An Ekman-type current, uniform in the horizontal but with a vertical shear, is also incorporated. The results indicate that the error in the current measurement is highly dependent on the sensor motion induced by waves. The error magnitude is proportional to the wave momentum or Stokes drift and depends on the wave development state and the wind-generated current’s magnitude. The error obtained in the current measurement is analyzed by considering that the buoy only responds to low-frequency waves up to a maximum frequency. That maximum frequency is referenced concerning the peak frequency of the third moment of the spectrum (i.e., the Stokes drift spectrum). It allows us to classify the current time average into three ranges with respect to the maximum frequency: (1) Eulerian average, (2) wave-following average, and (3) intermediate case of undulating average where results cannot be generalized. The measurement error is most important in the region above the wave troughs. However, the error is also considerable in the region confined below the wave troughs and down to the Stokes drift e-folding depth. The error is particularly relevant in conditions of developed and energetic waves (𝐻𝑠>3 m), where the surface Stokes drift can reach values above 0.1 m/s. It should be noted that measurement error can exceed the value of the Stokes drift at the sensor depth for certain mooring line motions. Those results should help better interpret in situ near-surface current measurements obtained from various devices.