We present the results of a spectroscopic follow-up of various puzzling variable objects detected in the OGLE-III Galactic disk and bulge fields. The sample includes mainly short-period multi-mode pulsating stars that could not have been unambiguously classified as either delta Sct or beta Cep type stars based on photometric data only, also stars with irregular fluctuations mimicking cataclysmic variables and stars with dusty shells, and periodic variables displaying brightenings in their light curves that last for more than half of the period. The obtained low-resolution spectra show that all observed short-period pulsators are of delta Sct type, the stars with irregular fluctuations are young stellar objects, and the objects with regular brightenings are A type stars or very likely Ap stars with strong magnetic field responsible for the presence of bright caps around magnetic poles on their surface. We also took spectra of objects designated OGLE-GD-DSCT-0058 and OGLE-GD-CEP-0013. An estimated effective temperature of 33 000 K in OGLE-GD-DSCT-0058 indicates that it cannot be a delta Sct type variable. This very short-period (0.01962 d) high-amplitude (0.24 mag in the I-band) object remains a mystery. It may represent a new class of variable stars. The spectrum of OGLE-GD-CEP-0013 confirms that this is a classical Cepheid despite a peculiar shape of its light curve. The presented results will help in proper classification of variable objects in the OGLE Galaxy Variability Survey.