If I counted right, there are 15 heroes who can deal 2 damage instances at the same time. These being:

1. Earthshaker (Aftershock + Fissure or Echo Slam)
2. Spectre (Desolate + attack damage)
3. Clinkz (Searing Arrows + attack damage)
4. Spirit Breaker (Greater Bash + Nether Strike or attack damage)
5. Lich (Frost Blast 2 damage instances on primary target)
6. Enchantress (Impetus + attack damage)
7. Silencer (Glaives of Wisdom + attack damage)
8. Elder Titan (Echo Stomp 2 damage instances)
9. Ancient Apparition (Chilling Touch + attack damage)
10. Nyx Assassin (Vendetta + attack damage)
11. Templar Assassin (Meld + attack damage)
12. Batrider (Sticky Napalm + any other damage from bat)
13. Outworld Devourer (Arcane Orb + attack damage)
14. Storm Spirit (Overload + attack damage)
15. Bounty Hunter (Shadow Walk + attack damage)

In the first 8 cases (Earthshaker to Elder Titan), when Abaddon gets hit by them when at low enough health, Borrowed Time triggers on one of the 2 damage instances and heals the second instance into a heal.

However, on the remaining 7 cases (Ancient Apparition to Bounty Hunter), although Borrowed Time still triggers on one of the 2 damage instances, it does not turn the second one into a heal. It completely bypasses Borrwed Time. So Abaddon can be killed by them during Borrowed Time.


Example: Let's take Nyx with Vendetta. Let's say Nyx has exactly 100 attack damage and Vendetta deals 300 damage for simplicity's sake. If Abaddon is at let's say 300 health and is hit by this Nyx, this happens:

1. Nyx's attack connects, applies 100 damage
2. Abaddon is now at 200 health, Borrowed Time auto-actives from that attack since he took damage while below 400 health. We are now under the effect of Borrowed Time.
3. The Vendetta damage gets applied. You'd think it would turn into a heal right? But it doesn't. It applies its 300 damage and kills you

Compare this to one of the first 8. Let's take Elder Titan's Stomp. Same scenario, but different result. Let's say the Stomp does 200 damage per instance for simplicity's sake. Our Abaddon has 300 hp.

1. Elder Titan casts Echo Stomp and hits Abaddon.
2. The first damage instance of Echo Stomp puts his hp to 100. Borrowed Time now auto-activates due to taking damage while below 400 hp. We are now under the effect of Borrowed Time.
3. The second damage instance of Echo Stomp applies. Unlike in the previous case, Borrowed Time does turn this one into a heal.
4. Abaddon is now at 300 health again.

In both cases, 2 damage instances are applied on the same server tick.

Here for the other bugged heroes. The common thing these spells have, they apply their own damage first, instead of applying their damage after the leading damage instance/attack.
Spoiler: 

Ancient Apparition
* Chilling Touch damage applies
* Borrowed Time activates
* Leading attack damage applies, damages through borrowed time

Nyx Assassin
* Vendetta damage applies
* Borrowed Time activates
* Leading attack damage applies, damages through borrowed time

Templar Assassin (Meld + attack damage)
* Meld damage applies
* Borrowed Time activates
* Leading attack damage applies, damages through borrowed time

Batrider (Sticky Napalm + any other damage from bat)
* Sticky Napalm damage applies
* Borrowed Time activates
* Napalm proccing damage applies, damages through borrowed time

Outworld Devourer (Arcane Orb + attack damage)
* Arcane Orb damage applies
* Borrowed Time activates
* Leading attack damage applies, damages through borrowed time

Storm Spirit (Overload + attack damage)
* Overload damage applies
* Borrowed Time activates
* Leading attack damage applies, damages through borrowed time

Bounty Hunter (Shadow Walk + attack damage)
* Shadow Walk damage applies
* Borrowed Time activates
* Leading attack damage applies, damages through borrowed time