Your teacher's analogy works.
Think of it this way: If you had one molecule of strong acid, like HCl, you would need one molecule of base, such as NaOH to neutralise it. If you had 1000 molecules of a strong acid like HCl, you would need 1000 molecules of a base like NaOH to neutralise it.
So now if you had 1 molecule of a weak acid like acetic acid, you will still need one molecule of base to neutralise it, and if you had 1000 molecules of acetic acid you will still need 1000 molecules of base to neutralise it.
Neutralise generally means you have equal amounts of acid and base.
Now the only difference between the strong acid and the weak acid is: with your strong acid like HCl, all 1000 of the H+ ions are already sitting in solution waiting around to be neutralised. In the acetic acid solution you may only have 50 out of the 1000 ions waiting to be neutralised, however as soon as those 50 are neutralised the acetic acid realises that there is a deficiency of H+ all of a sudden so another 50 ions are dissociated and then they get neutralised, and this continues on and on until the entire 1000 are used up.
So like your teachers bank analogy, it doesn't matter whether you neutralise all 1000 in one go or 50 at a time if you start with 1000 of either you will still have to neutralise 1000 by the end.