Saved might be a strong word, but I've helped some projects get delivered over time and over budget, when four previous teams had failed to deliver at all.
That really is a significant warning sign. If you are dealing with an organization that is currently failing at something it can mean almost anything. They don't have the skills? They trusted a 3rd party supplier? They have one bad leader?
If you are dealing with an organization that has failed repeatedly on the same task, then you are dealing with entrenched issues that you are not going to easily change. If they've decided in the past that this project can be done by 10 people in 3 months with continuously increasing scope then there's a good chance that's how it will run this time too.
As far as I can tell the key is to learn to be diplomatic but emphatic about saying no.
You can only be an asshole to coworkers occasionally, and you need to recharge your karma in the meantime, so being the guy who always says "no, that's too hard. It would take too long" will just mean you'll get sidelined and routed around.
You need to be the guy who says "Yes! That feature sounds awesome! Our customers will love it! Which feature can we cut to make time for it?"
Pretending that design and scope will stay the same for 6 months is fantasy, but as it changes you need to maintain a non-negotiable climate of people working regular hours and not getting bullied into making impossible promises.
Give somebody a goal 10% out of his comfort zone and he will work harder to meet it. Give somebody a goal 50% out of his comfort zone and he will make jokes about the whooshing sound that deadlines make as they fly past.