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Under net (PHB pg 148):
Dealing 5 slashing damage to the net (AC 10) also frees the creature without harming it, ending the effect and destroying the net.
So does the net need to take a total of 5 damage over any number of attacks, or does a single attack have to do 5 damage?
I would assume 5 cumulative damage, but I'm not sure if this is intended to be an exception to object health based on the wording.
PH doesn't say how much HP does the net have. What it does say, if you deal 5 slashing damage, you free the creature and destroy the net.
The specific details is up to the DM, but the net itself might have more than 5 hit points. For instance, "Fallen Net" trap from the DMG suggests the net has 20 hit points:
The net has AC 10 and 20 hit points. Dealing 5 slashing damage to the net (AC 10) destroys a 5-foot-square section of it, freeing any creature trapped in that section.
(DMG page 122)
The "Object Hit Points" table says the medium object can have 5-15 hit points, depending on its resilience. In this case, dealing 5 or more slashing damage with a single strike still destroys it. It's like the "Massive damage" rule for an object.
So, to get free, you can repeatedly damage the net until all its hit points expire, or deal 5 slashing damage with a single strike.
If you are the DM, and you assume 5 cumulative damage, that means the net itself has 5 hit points. It's perfectly normal — according to the "Object Hit Points" table, this is fine for a medium fragile object. So, this particular net just wasn't very strong.
The point is that damage has a type in D&D 5e. Slashing is a type of damage. You have to do slashing damage in order to cut the net.
While you have to cut the net, since it does not specify "all at once" then it is by default cumulative.
Here are the kinds of damage in this edition of the game:
Damage Types
Different attacks, damaging spells, and other harmful effects deal different types of damage. Damage types have no rules of their own, but other rules, such as damage resistance, rely on the types. The damage types follow, with examples to help a DM assign a damage type to a new effect. {types without explanation text}
Acid. Cold. FireForce. Lightning. Necrotic. Piercing. Poison.
Slashing. Psychic. Radiant. Thunder. (PHB . 196)
While D&D 5e is not highly simulationist, this is a nod to versimilitude.
The description doesn't specify that the damage has to be done all at once, so 5 cumulative damage would be my interpretation. I don't think it's been clarified in Sage Advice or anything.
The description says:
Dealing 5 slashing damage to the net (AC 10) also frees the creature without harming it, ending the effect and destroying the net.
Given that dealing that 5 damage destroys the net, it's functionally equivalent to an object's "HP".
The slashing damage being dealt has no effect on its HP. It is a specific condition for a specific effect. As such, 5 of another type of damage may not destroy the net. I couldn't find any specification on whether or not the 5 slashing must be dealt in one attack.
The Dungeon Master's Guide has some details regarding an objects hit points (table recreated):
...suggested hit points for fragile and resilient objects that are Large or smaller.
$$ \text{Object Hit Points} \\ \begin{array}{|c|c|} \hline \textbf{Size} & \textbf{Fragile} & \textbf{Resilient} \\ \hline \text{Tiny (bottle, lock)} & \text{2 (1d4)} & \text{5 (2d4)} \\ \hline \text{Small (chest, lute)} & \text{3 (1d6)} & \text{10 (3d6)} \\ \hline \text{Medium (barrel, chandelier)} & \text{4 (1d8)} & \text{18 (4d8)} \\ \hline \text{Large (cart, 10-ft.-by-10-ft. window)} & \text{5 (1d10)} & \text{27 (5d10)} \\ \hline \end{array} $$
I would consider the net at least a Medium object (since it has to restrain up to Large creatures) and nets would be designed to be durable making its Hit Points by these rules somewhere around 18 (4d8).
