Recent progress in the development of superconducting nanowire single-photon detectors (SNSPDs) has delivered excellent performance and has had a great impact on a range of research fields. The timing jitter, which denotes the temporal resolution of the detection, is a crucial parameter for many applications. Despite extensive work since their apparition, the lowest jitter achievable with SNSPDs is still not clear, and the origin of the intrinsic limits is not fully understood. Understanding its intrinsic behavior and limits is a mandatory step toward improvements. Here, we report our experimental study on the intrinsically-limited timing jitter in molybdenum silicide SNSPDs. We show that to reach intrinsic jitter, crucial properties such as the latching current and the kinetic inductance of the devices have to be understood. The dependence on the nanowire thickness and the energy dependence of the intrinsic jitter are quantified, and the origin of the limits is exhibited. System timing jitter of 6.0 ps at 532 nm and 10.6 ps at 1550 nm photon wavelength has been obtained.