To better understand the physical connections in sympathetic solar eruptions, we investigated the interaction between two nearby filaments and their successive partial eruptions in the active region (AR) NOAA 12866 on 2021 September 9 by using data from the Solar Dynamics Observatory and the New Vacuum Solar Telescope. Based on H alpha and extreme ultraviolet observations, we found that the right part of one filament (F1) became active first and experienced an obvious rolling motion. Then the whole body of the filament became wider and expanded toward another filament (F2). They collided with each other, and the interaction between them was accompanied by the brightening and bidirectional flows that appeared between them. This implies that magnetic reconnection occurred between the threads of two filaments. The interaction resulted in a rightward motion of F2 at first, and then its activation, and finally part of it erupted. Furthermore, when the erupted F2 deflected rapidly toward the middle part of F1, the left part of F1 erupted with its overlying magnetic fields pushed by F2. These observational results imply that these successive eruptions within a short time are physically linked, and this was caused by the interaction of the filaments. Nonlinear force-free field extrapolation reveals that the magnetic structure of the filament F1 was composed of several magnetic flux ropes with different twists. These results further advance our understanding of partial filament eruptions and sympathetic solar eruptions.