I discussed this here. All of the British destroyers sunk by aircraft at Dunkirk were hit by more than one bomb - though it's possible that, in some of these cases, the extra hits were just gilding the lily, and the first hit would have caused a sinking. In any case, none of them sank quickly, with most requiring multiple hits over multiple attacks to cause a sinking. There are cases where destroyers were sunk by single bomb hits - in 1942 alone, the destroyers Kipling and Lively were sunk by single bomb hits while operating off Egypt on the 11th May, while Maori was sunk by a single hit in Malta's Grand Harbour on the 12th February. Defender was sunk by a single bomb that didn't actually hit the ship, but exploded close alongside. Destroyers had no armour, relying on speed and manoeuvrability for protection. Most British destroyers, especially at the time of Dunkirk, had scanty AA armament. Damage control could save a heavily damaged ship, but RN training in this was spotty, especially in the early war. During the inter-war years, damage control training was only really given to a crew if the captain was an enthusiast for it.

Just as a note, I don't think the movie portrayed any destroyers being sunk by bomb. If I recall correctly (saw the movie on Saturday) there were only 2 vessels sunk by bomb, one being some sort of hospital ship and the other was a minelayer. The only destroyer I remember being sunk was sunk via torpedo (from a u-boat).

One point I haven't seen mentioned yet is explosives size. The destroyer that was sunk in the movie was hit by a torpedo, which contained much more explosives than bombs. The SC250 bomb, which was the standard general purpose bomb used by the Luftwaffe (and the Stukas depicted in the film) had about 130 kilograms of explosives whereas the G7a torpedo (I'm using this one as it left bubbles and we clearly see bubbles in the water in the movie) had 280Kg in its war head, just over twice as much.

Additionally, a bomb that hits the main deck and explodes inside the ship is going to produce different damage than an explosion that happens just outside the hull. A torpedo hit would generally open a massive hole to the interior unless there was armor and structures in place to absorb and mitigate the explosion (Battleship USS California sank at Pearl Harbor not because her "torpedo defense system" of armor and compartments were breaches, but because access ports to inspect said compartments were open for a planned inspection that morning and it allowed water to pass through and sink the ship). Destoyers are small and fast and pack a lot of offensive fire power, but at the expense of armor, so they can be fairly vulnerable to torpedoes.

Flooding can be contained completely or somewhat depending on damage control and circumstances, but in the movie the power immediately goes out, so any damage control parties would have been incredibly ineffective.