We present the analysis of 34 new VLT/X-Shooter spectra of young stellar objects in the Chamaeleon I star forming region, together with four more spectra of stars in Taurus and two in Chamaeleon II. The broad wavelength coverage and accurate flux calibration of our spectra allow us to estimate stellar and accretion parameters for our targets by fitting the photospheric and accretion continuum emission from the Balmer continuum down to 700 nm. The dependence of accretion with stellar properties for this sample is consistent with previous results from the literature. The accretion rates for transitional disks are consistent with those of full disks in the same region. The spread of mass accretion rates at any given stellar mass is found to be smaller than in many studies, but is larger than that derived in the Lupus clouds using similar data and techniques. Differences in the stellar mass range and in the environmental conditions between our sample and that of Lupus may account for the discrepancy in scatter between Chamaeleon I and Lupus. Complete samples in Chamaeleon I and Lupus are needed to determine whether the difference in scatter of accretion rates and the lack of evolutionary trends are robust to sample selection.