Deadlines are a reality in all businesses (Real deadlines or made up ones). But if you have a thing to create that takes 5-200 days what will you do?
You can pull a different estimate of 20-40 out of your behind when pressured to narrow it, and make the deadline 40 days. That might be a 50/50 hit/miss the deadline, with a worse case of running over to the full 200 day worse case (or even worse - even the 200 was a guesstimate). In some cases that's fine too. The decisionmaker might just want a date with 50% chance of hitting because it's better to land a sale and apologize for the delay, than to not land the sale and go bankrupt.
The correct course of action though is to not set the deadline in that meeting. The 5-200 day estimate is a way of explaining "Either don't take this on at all, or at least don't try to schedule anything urgent based on it".
In that case as a developer I'd suggest "Let me look at this for one afternoon, and I can return with an estimate that's 1/10th as large. Then let's meet again ad prioritize some work, schedule UAT and so on". PM's often look really surprised when they realize that all it takes might be to check if there is a functionality available already or if it needs to be implemented - which represented the difference between 5days and 200. If I go off and find it's already there, the estimate is 5-10, if I go off and find there is nothing and it has to be built from scratch, it's 150-200. But without that knowledge: 5-200.
Or, if a 20-200 day estimate is actually due to real implementation uncertainty, I'd suggest "This is potentially large and uncertain. Let's try to break this apart if necessary into something that has a 10-20 day estimate and some parts that have a longer horizon. Schedule the delivery/UAT of the first functionality in 20 days, and we'll by then have a good estimate of when the remaining work will be done".
If someone in my organization would try to convince me to pull a stupid estimate out of thin air, then pressure me to haircut that estimate, then schedule something based on it (such as a delivery), then pressure me to work overtime to meet it, then one of us would be leaving the org very soon.