Electrical insulation materials are highly exposed to electrical network-based electric and magnetic fields in power systems. In electrical fields, electrical insulation materials are prone to breakdown and cause failure. A transformer failure which is related to pressboard insulation may lead to total breakdown and hence system malfunction in a total manner. In this study, a test setup is used to conduct discharge tests for pressboards in different thicknesses where main interest is originated magnetic fields on the pressboards. These tests are fulfilled with spherical and rod electrodes in transformer oil where magnetic field sensors are used to acquire discharge-based magnetic field data. By investigating high-voltage stresses with different levels, possible breakdown voltage of a pressboard is predicted and statistically analyzed. In addition to magnetic field measurements, discharge current measurements are taken; however, contrary to conventional studies, this study assesses magnetic field data which are dependent on the thickness of pressboard insulation. For different voltage levels (13 kV and 22 kV for different stress levels), magnetic field measurements and current waveforms are obtained by using magnetic field sensors and high-speed oscilloscope. Magnetic field time series signals are subjected to wavelet analysis, and wavelet coefficients are obtained. Rather than utilizing time series current signals or time series magnetic field signals, wavelet coefficients of magnetic field signals are taken into consideration as a novel approach. These coefficients are processed by multifractal analysis, and finally, the integrity of the pressboard is determined as in proper mode (no failure) or in breakdown mode.