The properties of Alfvénic solar wind turbulence have been studied for decades using spacecraft measurements. In particular, the observation of spectral anisotropy of magnetic fluctuations has stimulated the development of several phenomenological models, one of the most popular being critical balance. However, the experimental validation of these models is intrinsically difficult because of the one-dimensional nature of the measurements provided by spacecraft instrumentation. In this work, a thorough search is performed in the Wind spacecraft database to extract samples of field-aligned fast solar wind, which allow the precise estimation of the parallel spectral properties of the magnetic fluctuations, and of their intermittency. Hilbert spectral analysis is used, in order to eliminate the possible role of nonstationarity and large-scale structures. Our results indicate that the spectral anisotropy predicted by the critical balance theory is not observed in the selected database, thus questioning the validity of the critical balance in the solar wind turbulence. A stochastic process characterized by a −5/3 spectral scaling, which is not necessarily attributed to usual turbulence, as indicated by the absence of intermittency, is indeed observed in the analyzed data samples.