Apparently, you have to deal 5 slashing damage at once
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 assume cumulative damage, it's actually not a problem
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.