The estimate everyone knew was wrong — how do you push back early?

Most late projects were late at approval. The duration was set by a date somebody needed rather than by the work, and the people who could see it said so quietly or not at all. Pushing back needs something more than an opinion. A range instead of a single number, a named assumption the estimate depends on, and a short list of what would have to be true for the optimistic case to hold — those are arguable in a way that "it feels tight" is not. What has actually worked for you, and at what point does it become the sponsor's risk to accept in writing?