The frequency-dependent viscous and elastic properties of fluids can be determined from measurements of the thermal fluctuations of a micron-sized particle trapped by optical tweezers. Finite bandwidth and other instrument limitations lead to systematic errors in measurement of the fluctuations. In this work, we numerically represented power spectra of bead position measurements as if collected by two different measurement devices: a quadrant photodiode, which measures the deflection of the trapping laser; and a high-speed camera, which images the trapped bead directly. We explored the effects of aliasing, camera blur, sampling frequency, and measurement time. By comparing the power spectrum, complex response function, and the complex shear modulus with the ideal values, we found that the viscous and elastic properties inferred from the data are affected by the instrument limitations of each device. We discuss how these systematic effects might affect experimental results from microrheology measurements and suggest approaches to reduce discrepancies.