This paper investigates instrumentation effects that may limit the usefulness of pre-compliance bench-top test methods for e.g. CISPR 25 radiated field measurements. Here the radiated fields are dominated by common-mode currents along the test cable bundle especially when the frequency is below several hundred MHz. From a known current distribution, fields can be computed. The currents can be measured using current probes with measurement instruments like oscilloscopes or spectrum analyzers (SAs). However, because of the low CISPR limits, very small currents (amplitude and phase) need to be measured. In this paper, as first step, the current values that lead to emissions at the CISPR limit are determined using method of moment (MoM) simulations. For analyzing the practical sensitivity of typical current measurement instruments, the noise figures of oscilloscopes and SAs are approximately measured and the compete measurement system is optimized with respect to probe selection and pre-amplification. It is shown that oscilloscopes and SAs achieve similar sensitivities, and both having noise figures in the range of 15-25 dB without pre-amplification. By setting corresponding fast Fourier transformation (FFT) parameters it is possible to obtain the same bandwidths in an oscilloscope as in a SA for the bandwidths typically used in electromagnetic interference (EMI) measurements. Further, oscilloscope-based phase measurement is explained. The noise induced errors are shown for the magnitude and phase indicating a lower minimal detectable signal of about -25 dBμA for less than 15° phase error.