Multiple hits poison

[Hilary]

Suppose you were struck by six poisoned dagger attacks all in the same round.

Would you really need to make 6 poison saving throws?

Or once you are attacked with poison in a given round, you are exempt from the effects of that particular poison until your next round?

How about 12 blowguns dealing 12 pi damage and 12d4 poison damage?

[PhoenixPhyre]

Suppose you were struck by six poisoned dagger attacks all in the same round.

Would you really need to make 6 poison saving throws?

Or once you are attacked with poison in a given round, you are exempt from the effects of that particular poison until your next round?

How about 12 blowguns dealing 12 pi damage and 12d4 poison damage?

Depends. Is it a save against poison damage? Then yes. Against only the poisoned condition? That doesn't stack. Against a mix? Yes. The condition doesn't stack, but the damage does.

[Kuulvheysoon]

It depends on the monster statblock. Some poison attacks, like the Flying Snake in the MM, don't require a save.

Bite. Melee Weapon Attack: +6 to hit, reach 5 ft., one target. Hit: 1 piercing damage plus 7 (3d4) poison damage.
On the other hand, if you were attacked by a half dozen Poisonous Snakes (also in the MM), you would need to make a half dozen saves against poison.

Bite. Melee Weapon Attack: +5 to hit, reach 5 ft., one target. Hit: 1 piercing damage, and the target must make a DC 10 Constitution saving throw, taking 5 (2d4) poison damage on a failed save, or half as much damage on a successful one.
Finally, there's the type of poison that applies the poisoned condition, like the Carrion Crawler.

Tentacles. Melee Weapon Attack: +8 to hit, reach 10 ft., one creature. Hit: 4 (1d4 + 2) poison damage, and the target must succeed on a DC 13 Constitution saving throw or be poisoned for 1 minute. Until this poison ends, the target is paralyzed. The target can repeat the saving throw at the end of each of its turns, ending the poison on itself on a success.
This one has no save against the poison damage itself, but you need to complete the save to avoid the poisoned condition. As soon as you fail one save against the poison, you're don't have to roll anymore saves against the poison. The condition won't stack (thankfully).

RAW, it shouldn't matter the source of the poison, but some DMs may make you roll separately for different applications/sources of the poisoned condition (I've seen it). Take the Hell Hound's poison, which has a distinctly different effect.
Bite. Melee Weapon Attack: +4 to hit, reach 5 ft., one target. Hit: 5 (1d6 + 2) piercing damage. If the target is a creature, it must succeed on a DC 12 Constitution saving throw against disease or become poisoned until the disease is cured. Every 24 hours that elapse, the creature must repeat the saving throw, reducing its hit point maximum by 5 (1d10) on a failure. This reduction lasts until the disease is cured. The creature dies if the disease reduces its hit point maximum to 0.Like here, for instance. I could see a DM asking you to make saves against Carrion Crawler poisoned and Hell Hound poisoned at the same time, for instance.