Classic bureaucracy calls for classic techniques; Radar on MAS*H knows what's up. Your best bet is to cultivate relationships that can help you get things done faster. The more important your project the easier this will be; from official resource allocation to a guy on the inside acting as a back channel. It may be worth working both ends of the spectrum at the same time.

If your project is important enough or the problem pervasive enough you might be able to get people together on an official solution. Raise awareness that your progress is being blocked frequently. An "all on the same team" approach to working on a solution will bring departments together to work out a path forward. This will often appear in the form of limited delegation of IT resource ownership, priority work queues, and dedicated/embedded resources.

If your project is NOT important enough and/or the problem lacks interest at the right levels(awareness raising fails) you might have to game the system. Get friends in the right places that can help you out when you have something blocking you. You might also have to just suck it up.