Out of interest, why didn't you counter the counteroffer at $140k? Realistically, you could've probably gotten $150-155k on the spot, right there. They basically just showed their hand telling they want/need you to stay.
If it's small you just have the CEO/founder who decides everyone's salary, if it's a big bigger you have a manager with a margin of negociation but can always get a signoff for a bigger raise by the CEO, and when it gets really big there is a hard grid with categories, ranks within the categories, lower bounds and upper bounds.
They won't tell you what this higher number is under any normal circumstances, and may lie about it, but it probably exists.
In face-to-face communications, those not accustomed to negotiating will often give in just based on misread cues. For example, if someone asks how much you want, and you say a number, stop talking until you get a response. Many beginner negotiators will interpret silence as disagreement (even a short amount of silence) and quickly add "but that number is negotiable".
I have this happen all the time when I'm doing profiles of new candidates, particularly when they are junior. They say "I"m looking for 60K, but I am willing to negotiate". I usually respond by telling them they have already backtracked before receiving any objections.
Practice or get some coaching, and until that time, try to do it via email.