The first possibility is that when you drill the head
off you are creating a big enough burr on the remaining shank that it is preventing you driving it out. I'd recommend you use a very sharp drill, run it at very low RPMs, and ensure you use lots of feed pressure on the drill. Lack of high feed pressure will allow the drill to work harden the rivet which will almost guarantee two things: a large burr and a dulled drill bit.
If you're cutting the head
off clean perhaps there is enough corrosion
between the rivet and the material to bind the rivet shank. I'd try Kroil penetrating oil
Good luck with it.