This is my experience with Gnome 3.18, it would seem to vary based on desktop environment, i.e. Nautilus does on Gnome and Dolphin does on KDE but not the other way round, best bet is use the file manager that came with your desktop.
With Nautilus, it will warn you first, here is the exact message
For Dolphin, the answer was no, it tried to copy a file larger than the destination, to the destination, it did not even warn me, just stopped as if it had finished, even left the broken file in the folder.
Thunar did warn me first, here is the message
PCman FM did not warn me and the same as Dolphin, just left the broken file on the drive, no error messages.