The requirement was straight forward to copy a 6.5 Gb file from Phone (I own a 64 GB One Plus Two) and when I tried to do the same - Android File Transfer refused !
The other options to transfer it via network were all slow and I did try shareit and Aid Droid but gave up very soon. I needed to transfer via the USB cable as wifi would take some time and I am not a very patient types.
Google showed me that adb might be able to help and it indeed did !
By default on my OS X El Capitan (version 10.11.1) - the adb was not in path. However I knew that it would be installed since I remember that I did have android studio installed. A little more google and it pointed that it would be under
~/Library/Android/sdk/platform-tools
By the way, I also added the path to my bash_profile just to make my life easier next time.
I ran the following command to copy the file from my sdcard0 to my mac. The option -p allows one to see the progress and I recommend the same if you wish to transfer huge files.
adb pull -p /storage/sdcard0/train_photos.tgz /.
That's it.
So now you know how to Bypass the Android File Transfer 4 GB Limit !
PS: You may need to verify that the adb is able to detect your phone and enable developer options on your phone.