All discussions about estimates are really just discussions about scope. I try to make that as clear as possible when I'm talking about time estimates. There is no 'negotiating' estimates. If someone is being a little pushy with trying to get a shorter estimate, I will always just rephrase it as a trade-off. 'Doing it in fewer than X months would mean dumping X feature/walking back on Y commitment,' whatever the case may be.

I think in general these conversations always go better when you make the other person feel like they have a choice. Really, in today's corporate world everyone is playing CYA at all times. The person trying to bring down the estimate is probably doing so because they are anticipating what their boss will say. By listing potential trade-offs for them, I'm helping them by giving them fodder for that talk that they'll be facing, and they'll be able to give their boss a choice.

You always get better reactions out of people when they have choices, rather than just presenting a stone wall to them. Having said that, some times the person will really just keep being pushy, and then fortitude is necessary. If you are someone who has trouble definitively saying 'no' when it's required, you're doing your team a disservice by being the person giving estimates. You really need to never slip with this; all it takes is for you to one time cave and say, "okay, maybe we can try to shave a week off" and now that person knows that your estimates are negotiable.

If you consistently phrase them as facts rather than bids (like the sort of estimate a contractor gives you for a house), you'll get better results.

You can tell when they are negotiating time instead of scope. It will be something like "surely we can do this in 2 days not 5?" Instead of "we probably will take 2-5 days to do this but need it in 2 so something of 3 days needs to come out somewhere else". The first one is saying 'work harder and just do it.' The second is 'ok it may take 2-5 days but lets pull something else out to make up for the deadline we want'.

I found orgs that negotiate time did not talk to their team in the first place. Then went on to promise something on behalf of the team. Now they do not want to eat the humble pie of 'I ran the numbers and you only get half' instead they get it going and just overbudget on the thing. Which just makes everyone in the process mad.

I remember an interview at a well known tech-ish company where the question started out as “Estimate this project” with a good description of the scope and team. After I said something like 4 months, interviewer said, “OK. The CEO says it needs to be done in 2 weeks. What now?”

I thought this was a basic project management question about trading off quality, scope, schedule, staffing, etc. and offered the usual suggestions of cutting scope (nope CEO demands entire scope), cutting quality (nope can’t compromise quality), increasing the team, contracting out, reprioritizing and realeasing P1s in 2 weeks (nope, nope, nope). Obviously I didn’t get the job.

Later I learned from an acquaintance inside the company that the right answer was probably “Work the team 24 hour days, nights and weekends, for 2weeks to deliver what the CEO wants. I think I dodged a bullet there.