The iron triangle of cost, scope, time, pick only two, is a model of project constraints. It's a good model, but sometimes confuses people who think they can only work on one dimension, while keeping the other two fixed. If the time and scope of the project are together so constraining that no amount of money can offset them, then your project won't succeed by raising the budget. If I give you a quatrillion of trillion dollars to put another man on the moon by tomorow afternoon, you won't be able to do it. And it's the same with software projects.
If you throw money at problems too late or expect to much from it, you still have the problems. If your project is late, and you bring in money too late, it won't save your project. Todd's answer covers this well so I won't insist on it any further. What I want to point out though is the fact that money is an enabler. Things that were "fixed" might all of a sudden not be fixed anymore.
For example, say your project consists of optimizing an application to run efficiently on small virtual machines, making use of hardware resources efficiently thus keeping cloud costs low and allowing for more efficient horizontal scalling in the future. You identified the work that needs to be done and you have two months to do it (for some reason). After one month the team dicovers that they will most likely miss the deadline. They can't reduce scope because all that's defined in there is needed for everything to work. And they can't move the deadline because that's fixed. So you bring in the money. Lots of it.
You can bring in more people but in doing so, you discover Brooks' law is 100% true, and now the team thinks they will be even later. But time and scope are still fixed. Or are they?
If you take the money and put it into the biggest virtual machies you can find, as many you can find, you no longer care about the scope of the project at all. Your fixed scope all of a sudden disappears entirely. It wasn't so fixed afterall. Your application can now be a resource hog on the cloud, and you don't care because you have money to pay the bills.
So in conclusion:
- it's not three constraints, pick just two. It's actually a balance of all and changes in one affect the others in some other way.
- more money can save your project but only if applied at the right time and on the right things.