I want to estimate conservatively, but project schedules don't allow for that.
Then your boss/team lead/project manager is doing it wrong. They need accurate information first. They may come back at a later point because they need to change the timing, but not letting you give them accurate information from the beginning doesn't bode well.
There's lots of methods out there to lead projects and software development. Let me focus on classic project management. To (over-) simplify, the idea is to get an accurate idea of how long each piece of the puzzle takes, what resources (time and materials), and what dependencies there are. One the project manager has that, they can chart out the critical path (the tasks that any delays will add time to the project). If that doesn't match what the stakeholders want, then something has to give. Commonly it could be more resources devoted to particular tasks, it could be lowering acceptable quality, it could be pushing out the timeline.
Really, if they've told you how long you have before you estimate it, they aren't doing their job of managing the project and instead are pushing it off onto you, without giving you the authority to fix more than your part of it. It's a recipe for missed deadlines. And it's all-too-common.
So you should be able to estimate conservatively, and then it's there call if they need that part faster. If they do, they need to be willing to put more resources towards it or accept a reduction in quality to get it done on time. Or to revamp the requirements. Or jigger other parts of the schedule so you can start sooner.
Other methods have entirely different ways to estimate. Planning Poker http://en.wikipedia.org/wiki/Planning_poker has adherents, and a big point behind that is that after listening to the story of a part, without discussions (and therefore influence), everyone puts out an estimation card and turns over at the same time. This allows everyone's estimate to be heard. From there you've got high and low estimates talking about why they think it will be long or short, and then go again until there is consensus. It's not trying to match a project plan, but come up with an accurate estimate in the first place.