The comments provided are not helping us provide specific suggestions mtnerd.
You said "The file that I tried to export/import was a one screen test with no images." ... what kind of file did you try to export / import and from where? Are you creating an aia file or something else? You said a one screen test ... was this just a part of your larger app? If you have multiple screens, you may have serious issues.
You said " So project/image size is not the source of the problem." OK but if your app has lots of layouts, especially embedded in one another, that design can ALSO create issues. However, we cannot see your code and you did not provide any blocks so who knows?
You said "The screen is linked to a Fusion table, but it would be very odd if that was the source of the problem." Why would you think that? Issues can occur anywhere.
You said "these projects need to have external data sources" ... what type of sources are they linking to? Web pages, Google Spreadsheets, DropBox or just to a Fusion Table?
Taifun provided a lot of general advice. Forum members are willing to help but you will have to provide more information.
Even though there are several app development competitions going on at the moment and lots of schools are finishing final projects, the load on the MIT build servers seems high but there is some free space. I . I just checked and will see if I can find someone at MIT. The probability your inability to build a project is something in your Project code. Just so we are straight, is the problem than you cannot compile the Project or that the compiled Project cannot link to your external data sources? When you link, how large and what type of file are you linking to
Regards,
Steve