Actually, with alternative splicing, one gene CAN make different proteins.
If you means different genes in the sense of a different sequence of DNA bases, then yes, different genes can make the same protein. That's thanks to the redundancy of the genetic code - look up the wobble hypothesis for more information.
However, entirely different genes - those at different loci or on different chromosomes - don't make the same protein, as far as I know (but what do I know?). This might be incorrect, since retrotransposons can make copies of themselves and insert these copies into different parts of the genome. Not sure if retrotransposons actually code for anything, though.