If an attack has Persistent 2 and a character is hit, and does not remove the status right away and it lasts say 2 rounds and then get hit again with say Persistent 3 from a different attack that ends up lasting 4 rounds, Does the character now suffer from Persistent 2 and then Persistent 3 at the start of the next round. So the next 2 rounds the character loses 5 Vigour, and the following two rounds loses 3 Vigour?
No. Persistent causes a number of damage dice, Combat Dice, according to the X given in Persistent X.
So Persistent 2 would cause 2[CD] of damage at the start of each turn following the round the character was hit by such an attack.
If a character was hit in round 1 be an attack with Persistent 2, which had 2 Effects on the damage roll, then this 2[CD] damage would by applied on round 2 and round 3. You roll the two damage dice each time anew, so that in one round it might come up with no damage at all (only blank faces were rolled), and another round might cause 4 damage (as two times 2 damage were rolled). If you roll any Effect on those damage dice, the Persistent 2 damage would continue another round per rolled Effect.
In the above example, if you got struck by Persistent 2 with 2 Effects in round 1, you would receive this damage in round 2 and round 3.
You would get the first 2[CD] damage in round 2. If this came up with 2 Effects, you would get 2 damage (as each Effect causes 1 damage vs. your Vigour), and the whole poisoning would continue 2 additional rounds, so up to round 5.
This way Persistent X can be quite deadly despite usually only rolling few damage dice, not enough to cause a Wound right away, as it whittles down your Vigour until it is at 0, when you get a Wound. And if you are at 0 Vigour and you get 1 point of damage or more, you get another Wound. And that continues until the Persistent X is no longer active, has run it’s course, or you are dead.
That is, why the Clear action is so very important if you didn’t buy off Persistent X for Doom in the first place (which is usually the better option).
Two hits by attacks with Persistent X are resolved separately, that means, the X damage dice are applied separately (not added up, so probably not causing a wound).
Example: If you got hit in round 1 by an attack with Persistent 2 with 2 Effects, this 2[CD] damage would apply in round 2 and 3, unless it gets prolonged by rolling Effects on the damage dice each of those rounds. If you get hit in round 2 by an attack with Persistent 3 with 4 Effects, that would apply 3[CD] damage in round 3, 4, 5 and 6, unless it rolls Effects and the damage is prolonged.
Round 1: hit by Persistent 2 attack with 2 Effects (this will usually cause Vigour damage in the first place, 2 Effects = 2 damage, maybe more if the attack’s other damage dice caused some not-null results, too)
Round 2: 2[CD] damage applied, coming up as 1 and 1+Effect, causing 2 damage against Vigour and prolonging the Persistent 2 damage up until round 4.
AND still in Round 2: hit by Persistent 3 attack with 4 Effects (causing at least 4 Vigour damage for the 4 Effects, maybe more for other damage dice results).
Round 3: 2[CD] damage applied, coming up as 1 and 1, causing 2 damage against Vigour.
AND still in Round 3: 3[CD] applied, coming up as 1, 2, 1+Effect, causing 4 damage against Vigour and prolonging the Persistent 3 damage up until round 7.
At this time, you have lost already 12 damage (or more, if some of the damage dice of the two initial attacks had a 1 or 2 result). That might be sufficient to cause your Vigour to drop to 0, causing a Wound, making the Clear action much harder, and setting you up for the Death spiral.
Round 4: 2[CD] damage applied, coming up as 1 and 0, causing 1 damage against Vigour, which was now already at 0, causing the 2. Wound.
AND still in Round 4: 3[CD] applied, coming up as 1, 2, 2, causing 5 damage against Vigour and as Vigour was at 0, causing the 3. Wound for receiving any damage while at Vigour 0, and causing the 4. Wound, because it caused 5 or more points of damage in one go.
In Round 5 you will be dead.
Persistent 1 is bad, Persistent 2 is deadly, Persistent 3 is simply a KILLER.
I recommend being very careful about the actual number X in Persistent X. If a player does not buy off this ongoing damage or has a weak Resistance in the first place, so not good a chance to succeed at the Clear action, the character will easily die.