An analysis of the sources of uncertainty in measuring the angular elements of the geomagnetic field with the D/I fluxgate theodolite on the basis of the Guide to the Expression of Uncertainty in Measurement published by the International Organization for Standardization is presented. Along with the uncertainty associated with random effects, the habitual measurement procedure evidences the existence of systematic effects that are often ignored in the daily observatory practice. Special emphasis has been put on the development of a plausible theoretical scheme to explain the origin of such effects, and a series of procedures are proposed to find their actual sources as well as several recommendations with the final aim to improve the accuracy of the observations. Other effects, which strictly do not seem to suit the traditional classification into either systematic or random, are also analysed. Some of the results obtained have been applied to the absolute instruments in use at the Livingston Island Geomagnetic Observatory and at certain European observatories. Systematic contributions to uncertainty are difficult to outline in a general case, since they depend on each particular instrument. On the other hand, an accurate estimation of the uncertainty associated with random effects has been obtained, concluding that their magnitude does not generally exceed 0.1 arcmin for an experienced observer, for either declination or inclination.