Is there a "Counter-Check"?

I have always wondered whether there is such a thing like a counter-check. I guess you all know the typical joke of desperation when you are in check(mate).. "can't I counter check you?"... Unfortunately you can't. And yet I was watching that film the other day "The President's Man". In one scene they were playing chess and immediately after a check the opponent checkmated the other one. I thought that the movie director must have had no idea of chess BUT! There IS counter-check and I just had one in a chess game! With the difference that the king of course, does not remain in check but a single move can cover up the king and check the opponent. I am still in the middle of a very interesting game (the one with the counter check I think!) and I will post it as soon as it ends.