Hey there,
I'm not sure who you bank with and each bank operates differently but NSF generally means the same across board = no sufficient funds which means your balance is 0 and below. I'm not sure of the circumstances as I've been both blessed as well as worked my arse off to not be in that position, however if I didn't have luck on my side, I probably would have an account that NSF (so far I'm knocking out 0 in the last ten years *knockonwood*) but basically I'm not judging and if this comes off like that, I apologize..I'm pretty blunt.
But bank accounts are in essence, something that the bank privileges you with, just like you privilege them with your money (to sit there and garner interest).
There is an agreement that you sign on for, and disclosures they provide you, much like a cc.
If you spend more than what you owe, you get dinged for that. If you don't meet certain criteria, you get dinged on that (monthly service fees). People often don't understand why there is one...well it's like a storage unit, you don't give those folks your stuff to hold for free do you? Just like your money, you don't give it to an institution to hold for free. They often make it complimentary because they want to win more of your relationship and they've done their part to try and ensure that (eg direct deposits of X waives the monthly service fee)
Some banks will wait for a few months before charging off your account if it's in the red. Some won't...it really depends on what they've experienced in the past and feedback...I can't tell you how many folks have themselves in a hole of -X,xxx but need more than a month to pay that down...
That's why your bank didn't close your account right away. If you wanted it closed, you should ask them to do so.
Also if the NSF has been paid off, depending on their procedures, most likely if it's small enough they'll let it go...if not, they can put you in chexsystem, and or take further action