In most countries, depending on severity of the event, the agency responsible for crash investigation can delegate investigation of the event to another entity.
Mind you, this is usually done for incidents, not accidents. However, sometimes an accident is clearly due to illegal operation, and sometimes that means that a) matter is passed directly to prosecution b) investigation is closed without conclusion due to explicit disregard of safety mechanisms, thus making further investigation useless to the purpose of aircraft accident investigation (under common rules from ICAO that NTSB also operates when it comes to aircraft)
Mind you, this is usually done for incidents, not accidents. However, sometimes an accident is clearly due to illegal operation, and sometimes that means that a) matter is passed directly to prosecution b) investigation is closed without conclusion due to explicit disregard of safety mechanisms, thus making further investigation useless to the purpose of aircraft accident investigation (under common rules from ICAO that NTSB also operates when it comes to aircraft)