I have this sort of discussion, at minimum, three times a week (we're in-house developers). Very often the project scope isn't even defined and upstream dependencies are known to be critically lacking and will need to be augmented to even perform the work. Yet we're always pressured for dates and the caveats given are immediately forgotten and almost never communicated.

The biggest pet peeve is when I give an estimate such as "I don't currently see a reason why we can't deliver X by the end of next week, assuming no other priorities come up." Of course something more important comes up in the mean time and I get scolded in a large meeting with a statement such as "You promised me I'd have X by Friday!" — this is especially infuriating when it is the PM's other project that is the higher priority interruption.

It might not be viable, but I used to "publish" internally my own project one pagers, and update the timelines - any discrepancies simply became a shield for this future arguments.

In short it sounds like you don't have the power to say no - but actually you do.

It doesn't get a lot of love on HN, but these problems are addressed by Scrum. Sprints have a pre-allocated amount of work in them, which makes estimating easier. You can't predict exactly how much time you'll lose through meetings and emergencies, but estimating in story points will add a rough statistical estimate.

Of course, if you already have a toxic development culture like the one you describe, then they'd probably abuse more agile processes too.

Instead of caveats, add to the estimate. Every caveat is some additional time.

Another thing that helped us is to answer with ranges instead of dates e.g., "this will take 1-3 weeks".