The triple constraint, in many cases, is a given to project managers. Regardless of approach, methodology, or industry, most PMs are not involved, or involved enough, in the early stages of the budget to properly set it. And although we all know that a business case estimate is a ROM (Rough Order of Magnitude) which means that the project will most likely be -50%/+200% of this ROM when completed, somehow that is forgotten and the next thing you know is that you have a fixed budget (and in many cases expectations or scope and time). Welcome to my world times 3!
What to do? Depending on your organization some aspects of the triple constraint may have some flexibility. Budgets may have some leeway. Maybe a management contingency was kept at the project or portfolio level that will provide some breathing room. Or maybe the scope is not as firm as it seems. Or maybe there’s more time (but that typically means more funds so that may or may not work). We, as PMs, need to probe to see what gives.
Don’t say “yes, sir/madam!” and hope for the best. All you are doing is delaying the inevitable. Face it now for a smaller cost than facing it later. Work with your manager to make sure he/she understands the reality and see what he/she suggests. They’ve probably been around the organization longer than you have and may have options.
Do sharpen your pencil and see what is not absolutely necessary. And it is possible that a feature does not need to be as complicated as originally designed. Finally, if you are using Agile, you may deliver enough value before you run out of budget that the organization is happy to call it done and consider it successful. If not enough value, they may be more amenable to finding more budget.