We present medium resolution near-infrared spectroscopic observations of 41 obscured and intermediate class active galactic nuclei (AGN; type 2, 1.9 and 1.8; AGN2) with redshift z ≲ 0.1, selected from the $Swift$/Burst Alert Telescope 70-month catalogue. The observations have been carried out in the framework of a systematic study of the AGN2 near-infrared spectral properties and have been executed using Infrared Spectrometer And Array Camera/VLT, X-shooter/VLT and LUCI/LBT, reaching an average S/N ratio of ∼30 per resolution element. For those objects observed with X-shooter, we also obtained simultaneous optical and UV spectroscopy. We have identified a component from the broad line region in 13 out of 41 AGN2, with full width at half-maximum (FWHM) > 800 km s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. We have verified that the detection of the broad line region components does not significantly depend on selection effects due to the quality of the spectra, the X-ray or near-infrared fluxes, the orientation angle of the host galaxy or the hydrogen column density measured in the X-ray band. The average broad line region components found in AGN2 has a significantly (a factor 2) smaller FWHM if compared with a control sample of type 1 AGN.