To set your mind at ease, reliable data transfer protocols incorporate error detection and retransmission. This ensures a perfect copy is made over the Internet, on your personal computer, or for whatever else. The only times this will fail is if the original data is corrupted or something is broken, perhaps due to a physical failure, in which case you should be informed of this. For example your computer would tell you there was a disk error, and the copy attempt would fail so you know it did not work.
Furthermore, this checking for errors happens along the entire chain, often layered on top of each other. For example if you wanted to copy a file on your personal computer's hard disk to a friend's computer across the country, there will be checks when the file is read from your hard disk disk, when the data is packaged for transmission, when it is sent over your local network, when it is sent over the Internet, when it is sent over your friend's network, when the data is unpackaged, and when the file is saved to your friend's computer's hard disk.
Reliable data transfer is how we control the Mars rovers, medical devices, cars, etc.