For newly developed measuring systems, it is easy to estimate type-B uncertainties based on technical data from the measuring modules applied. However, it is difficult to estimate A-type uncertainties due to the unknown type and level of interferences infiltrating the measuring system. This is a particularly important problem for measurements carried out in the presence of typical of power grid disturbances. The aim of the research was to develop a method and a measurement stand for experimental assessment of uncertainties in a measuring system that makes use of data acquisition modules containing analog-to-digital converters (ADCs). The paper describes, in detail, the design of a completed test stand. It presents an original application in the LabVIEW environment, which enables testing the dependence of the uncertainties with the quantity of the measurements averaged in a series, for different kinds and levels of interferences infiltrating the measuring path. The results of tests for several popular measuring modules are presented. An analysis of the determined uncertainties was carried out in relation to the parameters of the tested measurement modules and for various levels of interferences. It is proved that an increase in the number of averaged measurements to approx. 100–200 always results in a decrease in uncertainty for each tested module and under all conditions. However, a further increase in the quantity of measurements, even up to 1000 averaged measurements, proved reasonable only for high-accuracy modules, in particular with a high level of interferences. An excessive increase in the quantity of averaged measurements proved a low effect for modules characterised by a low resolution and with a low level of interferences. The measurement results also proved that when estimated, uncertainties in the interference probability distribution are significant, especially if they deviate from normal distribution.