Yes you can... though I know this is not possible on a file system such as FAT, it is possible on NTFS, not to mention many other file systems. I've done this many times while downloading on old and modern versions of the MacOS. The reason why this is possible is because once a file is opened for reading or writing, the location specified in the directory tree is irrelevant, since you get a pointer to the file instead.
Take a look at mIRC's file handling commands. You specify a handle name rather than the file's path each time you read and write.
It is not possible in Windows. Try downloading a file in Chrome, Firefox, or mIRC (DCC), and try to move the file. It will give you an error message.
