Hi Yuri,
it depends on where is the file. If the file resides on the internal sdcard, you can't change the permissions (also if you are root) due to security limitations. The internal sdcard is mounted as FUSE file system and it manages permissions for the files stored. However, if the file resides in the private app folder (i.e., /data/data/app_name/files) the chmod should work.
Furthermore, let me say that put 777 for a file is never a good idea :) If you need to share the file among apps maybe you can consider the use of a ContentProvider.
hope this help.
Simone