FCPX 10.6.5 file export problem. After exporting the file fails to play at approx the same point every time.

In addition to the always excellent advice from Luis, it might also be a corrupt render file. To clear the render cache, select the library in the FCP browser, choose File->Delete Generated Library Files... Choose All Render files.

Manage render files in Final Cut Pro for Mac - Apple Support 

how much free space on the system and media drive? What are your export settings?

When the application misbehaves, it often helps to delete its preferences. To delete preferences, hold down command and option while starting FCP X and click Delete Preferences. The application will start in a default state, and open an “Untitled” library.

Reopen the library you were working on, set the preferences the way you like them, and try again.