impossible scores

My brother and I were talking about impossible hands to get. We have always known that 19 is an impossible hand to get. It has even been passed down in our family that if you get a 0 when counting a hand that you got a 19. But we were trying to figure out if a 25, 26, or 27 was possible. Can any one clear this up for us and if you can get them, how?

25, 26 and 27

Yes, 25, 26 and 27 are all impossible cribbage scores, along with 19!

It might be interesting to try and prove it though!