The next time you receive one of these assignments, review the tasks that are involved in completing it to your boss' satisfaction. Then, break down the assignment into what I call Proposal A and Proposal B.
In Proposal A, offer a timeline that includes the money necessary to obtain the resources you need to complete the assignment to your boss' specifications. List those resources and the reasons you need them as precisely as possible. In Proposal B, supply a similar timeline, only without the additional resources that will aid in its successful completion. Indicate any areas that may be lacking as a result of the missing resources.
When you're finished with both proposals, present them to your boss. While acknowledging that Proposal A may cost more, you can point out the resources to ensure the job is completed more efficiently and effectively. When you discuss Proposal B, stress that the lack of resources means the lowered cost will be balanced by the additional time required to complete it.
After reviewing your two options, the ball will be in your boss's court. If he or she chooses Proposal A, you now have the explicit go-ahead to get the job done fast and well. If the choice is Proposal B, you have your boss's acceptance that it will take longer to complete. In both cases, you've set yourself up to flourish instead of flop.