If the project managers are giving you unrealistic deadlines given your workload, then push back or escalate. Part of a project manager's job is to manage the expectations of the project's customers. If they are not doing that they aren't doing their job.
It is part of a project manager's job to manage expectations of the clients in line with the available resources to do the work. Allowing them to low-ball the allocation does you no favours either way. If you accept low-balled schedules you will be perceived as not meeting expectations, so the perception of your performance will suffer. If you push back you may be perceived as uncooperative. To a greater or lesser degree the conflict is unavoidable, so there is relatively little to lose by being assertive about your time.
In the world of the PM, delivery is everything. Most of the PM's job in a corporate environment is trying to get resources to do the job, deliver with insufficient resources and get people who they have no direct authority over to do stuff needed for the project. The archetypal project manager is a bully (sorry to any PM's reading this) but this is a necessary skill to manage projects in any environment with incumbent politics and responsibility-without-authority. Much of the PM's work is endless pushing and prodding to get disinterested third parties do to things they don't really want to do.
You can push back. Project plans are flexible and expectations can be managed. Much of the time the PM is just trying it on. Stand firm about your time and be prepared to admit that you cocked up your estimates if you do get it wrong. If it took longer than you said, be up front with it.
If you have to involve management in prioritising the work, make sure that there is a paper trail of the priorities and make sure it gets circulated to all PM's and stakeholders involved. You are the pig and they are chickens.