This was an issue that I spent a good amount of time working with a customer on; when you add the Project Server web-parts to other SharePoint sites in other web applications, in some cases the Project Center / Project Details webpart will give the following error always:
"Project Center cannot access the project(s) you are trying to view. It is most likely that you either don’t have permissions to view the project, another user has deleted this project(s) before you were able to view it or that another user is in the process of publishing the project."
Quick note: The webpart can be added to other sites using the steps discussed here by Christophe.
After a long investigation the problem was found to be a repeatable issue caused by the use of Host Headers in the Alternate Access Mapping configuration of the SharePoint farm.
In short; when the PSI URL configured in the .webpart XML web part configuration file and / or the site in which the webpart has been added uses a host header other than the NETBIOS name, then this problem will occur.
All other Project webparts work without issue, Reminders, My Tasks, etc although I have not tried them all, I might guess that the Resource Center webpart would have the same issue.
Eventually we confirmed with Microsoft support that this issue is a bug and no hotfix is currently available.
In terms of workarounds, not many good ones exist. I tested adding a second Alternate Access Mapping pointing to the NetBios name, however this only works if all clients use the non-hostheader address, in other words:
Site: "Project XYZ Workspace" accessed via http://servername/site/projectxyz will work BUT when accessed via http://hostheader/site/projectxyz (which is the same site) it DOES NOT WORK. That obviously has major drawbacks.
In the end in my case our solution was to move the PWA site into the other web application.
I hope this helps someone out there.