Is it necessary to read every single byte to check if a copied file is identical to the original?

Question

I recently learned of a program called Total Commander. It's a Windows Explorer replacement and has its own stuff to copy files. To check whether the files are identical, instead of calculation a CRC, it literally checks every single byte, one at a time, on both the original and the copy.

My question is: Is this necessary? Can CRC or any other such technique go wrong? Should you, as a programmer, try and implement this perfect but slow system, or is it too extreme?