What if your project is late because some managers or seniors fail to cooperate?

Without encouraging "future career prospects" I can only make suggestions about what you as a project manager can do the next time to cover your own posterior. (None of these suggestions address the behavior of the managers since this appears to be outside the scope of your authority and control as the project manager. Without the organizational authority to affect their behavior, there is little point in trying to impact them. You probably won't succeed and it may have serious repercussions for you personally.)

1. Don't take the situation too personally. There is a real danger in getting too emotionally "involved" in your projects. When this happens, anything that negatively impacts the project - whether you can do anything about it or not - takes on a sinister aspect. You must accept that there will always be things that will impact your projects over which you have little or no control. When these occur, you can only react as best you can with the good of the project as your primary aim.

2. Make sure that the impact of withheld information, resources, work output, etc., is clear. A good change-control process is helpful here. It allows you to describe the change being made as well as the impact of that change on the project. Document this and be sure that everyone who should be informed is informed.

3. Realize that shifting priorities are a fact of organizational life. Priorities change constantly in any organization. New challenges arise that require a response from the organization and that response requires that resources be moved from one activity to another. In most instances, those resources come from projects that are as a result of the shift in emphasis no longer as important as they were yesterday. Unfortunately, many times, the project manager is not told the reason they've lost their resources.

4. Document what happens. Always document the things that happen during a project. Never assume that "everyone knows why this happened." They may, but, then again, they may not, or they may have a completely different understanding of the situation. Try to document the occurrence in a factual way. Try to avoid accusations and conjecture about "why" the thing happened. Document what happened and the impact it had on the project. A good change-control system can help with this. This documentation should become part of the total project documentation and can be included as part of the final project report. A good, carefully worded narrative about why the project was delivered late can reference this documentation.
5. Use your sponsor or product champion. A sponsor is someone in a position of authority in the organization who has agreed to act on behalf of you and the project when an issue is outside your scope of authority and control. If you do not normally identify a sponsor for your projects, seriously consider doing so. One of the functions of a sponsor is to intercede in situations like the one described. When a conflict occurs, the sponsor should be informed and asked for both advice and for direct assistance in resolving the conflict. The most common conflicts are over needed resources but they can also occur over issues of cooperation and delivery of work or information.