We present deep imaging observations, orbital dynamics, and dust-tail model analyses of the double-component asteroid P/2016 J1 (J1-A and J1-B). The observations were acquired at the Gran Telescopio Canarias (GTC) and the Canada–France–Hawaii Telescope (CFHT) from mid-March to late July of 2016. A statistical analysis of backward-in-time integrations of the orbits of a large sample of clone objects of P/2016 J1-A and J1-B shows that the minimum separation between them occurred most likely ∼2300 days prior to the current perihelion passage, i.e., during the previous orbit near perihelion. This closest approach was probably linked to a fragmentation event of their parent body. Monte Carlo dust-tail models show that those two components became active simultaneously ∼250 days before the current perihelion, with comparable maximum loss rates of ∼0.7 and ∼0.5 kg s‑1, and total ejected masses of 8 × 106 and 6 × 106 kg for fragments J1-A and J1-B, respectively. Consequently, the fragmentation event and the present dust activity are unrelated. The simultaneous activation times of the two components and the fact that the activity lasted 6–9 months or longer, strongly indicate ice sublimation as the most likely mechanism involved in the dust emission process.