A statistical analysis of fine splitting of C IV, N V, O VI, Mg II, Al III and Si IV doublet absorption lines in quasar spectra is carried out in order to estimate a possible time variation of the fine-structure constant {alpha}= e^2^(h/2{pi})c (~1/137) over cosmological time scales t~10^10^yr. The observational basis of the analysis is a catalogue of 1414 pairs of wavelengths with redshifts z = 0.2 - 3.7, compiled from data published in 1980-1992. Robust statistical estimates like the ``trimmed mean'' are used as well as the least squares. No statistically significant time variation of {alpha} is found. The estimate {alpha}^-1^d{alpha}/dz = (-0.6+/-2.8)10^-4^ is obtained. For the 95% significance level, an upper bound on the rate of a relative variation of the fine-structure constant is |{alpha}^-1^d{alpha}/dz| < 5.6x10^-4^, which corresponds approximately to |{alpha}^-1^d{alpha}/dt| < 4x10^-14^ yr^-1^. This limit represents the strongest up-to-date restriction on the possible time variation of {alpha} for the epoch 0.2 <= z <~ 4.