SOLVED! - Cannot Copy Big Files to External HDD

The chance is that your external hard disk is formatted with FAT32 file system, which has a file size limit of 4GB. One workaround is to re-format your external HDD using NTFS file system, which has a maximum file size limit of 16EB.

By default, most external hard disk drives are pre-formatted with FAT32. This file system is chosen because of its wide compatibility with most operating systems, including Mac OS X and Linux.

The down side of formatting your external har disk with NTFS is that, if you had a Mac OS X computer, you may read but cannot write on it. In fact, Mac OS supports NTFS read/write. However, this capability is not enabled by default. If you want to enable it, surely you can google and will find articles on it. However, those who had tried write to NTFS disk on Mac OS found its support buggy and corrupting files...

My suggestion, keep two external hard disk drives, one with NTFS and the other with FAT32, or even the native Mac HFS file system, if files bigger than 4GB has to be stored.