Rims of solar filaments often appear brighter than the background chromosphere, but their physical nature is still poorly known. Last year, we observed a filament with a bright rim. The rim was bright in H alpha but not in Ca II 8542 line. Using the cloud model, we inferred physical parameters of the region from the spectral profiles. As a result, we found that the Doppler width of the H alpha line is very large, which implies temperature as high as 50000K. In addition, the value of the source function of the H alpha line is 0.7 times the continuum intensity of background profile. These results suggest that the bright rims might be a region of intense heating, probably associated with a current sheet. To further investigate this possibility, we carried out more observations this summer. We will present new results obtained from the analysis of these observations and discuss the physical implication of these measurements on the nature of bright rims and the filaments.