Frequency-domain operational vibration responses have been widely applied in structural health monitoring (SHM). Characterizing and quantifying their uncertainties are of fundamental importance for enhancing the robustness of SHM technologies. The classic complex Gaussian distribution is being increasingly used to model the distributions of fast Fourier transform (FFT) coefficients due to its elegant and convenient mathematical nature. However, the field-test data analysis of engineering structures under operational vibrations in this study emphasize the possibility of non-Gaussianity of some observations. The higher peaks and heavier tails than those of a Gaussian distribution emerge as observable features. As a member of the general family of elliptically symmetric distributions, the t distribution has been widely recognized as a useful extension of the Gaussian distribution for the robust statistical modeling of data sets with heavier-than-normal tails. In this paper, we consider using the complex-valued t distribution to characterize the FFT coefficients with high kurtosis and heavy tails. The probability density function (PDF) of multivariate proper complex t random variables are derived based on the equivalent counterparts in the real-valued domain. The marginal PDFs of the real part, the imaginary part, the magnitude, and the phase of a univariate complex-valued t random variable are also derived analytically based on advanced integration techniques. The field test data of different engineering structures provide an illustration of the performance of complex Gaussian and complex t probabilistic models at different frequencies evaluated via the K-S test, goodness-of-fit test, and probability plots.