I think this is answered by the rules on Saving Throws:
Glossary wrote:
When to Make Saving Throws
[...]
If a creature is subject to an effect that a save can end, the creature makes a saving throw against that effect at the end of each of its turns. An effect that a save can end includes one of the following notations: “save ends,” “save ends both,” or “save ends all.”
When two effects are followed by the notation “save ends both,” they are treated as a single effect when making saving throws against them. For instance, if a creature is “weakened and slowed (save ends both),” the creature makes a single saving throw against those two conditions whenever the time comes for it to make saving throws. Similarly, effects that are followed by the notation “save ends all” are treated as a single effect when saving throws are made against them. However, if a creature is subject to separate “weakened (save ends)” and “slowed (save ends)” effects, it must save against each separately. When a creature makes saving throws against multiple effects that a save can end, its player chooses the order of the saving throws.
Sometimes a power or other game feature allows a character to make a saving throw immediately against an effect that a save can end. If the character is allowed to make a single saving throw but is subject to multiple effects that a save can end, the player chooses which of those effects to make the saving throw against.
Identical Effects that a Save Can End
If a character is subject to identical effects that a save can end, ignore all but one of those effects. For instance, if the character is dazed (save ends) and then is attacked and again becomes dazed (save ends), ignore the second effect, since it is identical to the first one.
Identical effects never require multiple saving throws. A creature does make separate saving throws against effects that aren't identical, even if they contain the same condition. For instance, “dazed (save ends)” and “dazed and immobilized (save ends both)” are not identical effects, so separate saving throws are made against each of them.
This is how I read it:
Character A is suffering the effect <ongoing 10 poison damage, save ends>.
Monster B uses a power on A that applies <ongoing 10 poison damage, save ends>.
Since that effect is identical to one Character A is already suffering, it's discarded instead of being applied.
B then uses a power on A that applies <ongoing 5 poison damage, save ends>.
Since that is not identical to the effect A is already suffering, it is applied.
However, since the ongoing damage has the same type as the preexisting effect, A only takes ongoing damage from the higher of the two.
Thus, A makes saves against the two effects separately, and if he successfully removes the original effect without removing the new one, he will then start taking 5 ongoing damage instead of 10.
EDIT: Rereading the section on Ongoing Damage, I've become uncertain:
Glossary wrote:
Different Types of Ongoing Damage: If a creature is subjected to ongoing damage of different types (including no type) at the same time, it takes damage of each of those types every round, and it must make a separate saving throw against each damage type.
The bolded section makes it sound as though one saving throw clears all instances of ongoing damage of a given type, in which case my conclusion above is wrong and yours is correct. I've posted the question in the 4e Discord and forwarded it to an acquaintance who is something of an expert. I'll let you know if they offer any further insight.
EDIT2: On the other hand, there's the case of effects that impose both ongoing damage and a second condition on a save-ends duration. The rules are clear that non-identical effects stack even if they contain conditions the target is already suffering, so, for example:
Character A is suffering the effect <ongoing 10 poison damage, save ends>.
Monster B uses a power on A that applies <ongoing 10 poison damage + slowed, save ends both>.
Since the effect is not identical to the one A is already suffering, it is applied.
A must save against each effect separately; a successful save against one does not end the other even though they share the <ongoing 10 poison damage> condition.
This is my current conclusion:
- For effects that only apply ongoing damage of a given type, the creature can only suffer instance at a time.
- If the ongoing damage of the new effect is greater than what is being currently suffered, it overwrites the previous effect.
- If the ongoing damage of the new effect is lesser than what is being currently suffered, it is discarded.
- If the effect imposes both ongoing damage and another condition, then it can coexist with another effect that imposes ongoing damage of the same type, but the sufferer only takes damage equal to the highest ongoing damage number per type.