One of the primary responsibilities of a project manager is to build a project schedule and assign activities from the schedule to team members for execution. After you assign work to team members you should hold them accountable for having the work completed within expectations.
On the surface this sounds very cut and dry. But is it really fair? If a team member didn't have any input into the estimating process to begin with, is it fair to hold them accountable for the work? The answer is no.
There are two ways to make sure that the team members buy into the estimates for schedule and budget. One way is to see if you can get the team members involved with the estimating process upfront. This is not always practical, but sometimes it's possible. In fact, you may need the help of the project team members to actually create the budget and schedule to begin with.
On the other hand, on many projects the project team members are not assigned until the project schedule is already in place. The people that created the initial estimates have to make some assumptions about "average" team member performance and make estimates based on those assumptions.
In this case it is appropriate to assign work to a team member and ask him to validate whether the estimate seems reasonable. You're not looking for an estimate with a 100% confidence factor. You're just trying to validate whether the estimate for schedule, effort and budget seem reasonable. If the team member says yes, then you can hold him accountable for completing the work within those estimates.
Of course, when you first assign the work, the team member may not know enough to say whether the estimate is reasonable or not. It may take a little time and he may actually have to start working on the activity. This is fine as well. In this case, you assign the work, along with the estimated effort, budget, and deadline. You then ask the team member to validate whether the work can be accomplished within these estimates. If he feels that the estimates are incorrect, he needs to communicate this to you AS SOON AS POSSIBLE. He is then obligated to provide a more realistic estimate.
The project manager can push back to validate that the new estimate is more accurate than the old one. Some negotiation may occur. However, when the project manager and team member are in agreement, then the project manager can hold the team member accountable.
The key in this scenario is that the team member must notify the project manager that the prior deadline is unachievable as soon as he realizes it. He can't wait until the original deadline is missed and then say that the estimates were off.
These approaches are the way to hold team members accountable for the work, while also being fair to them by allowing them to have input and buy-in to the estimates.