Wafer thin dies like Nesties and Framelits will cut through cereal box but that's about their limit. If your board is only a little thicker then it's worth a try, you might find it will work. The worst that can happen is that your die only goes part way through your board and you don't get a completely cut shape so you've only lost a bit of board. Just make sure you're not forcing something too thick through the machine by adjusting the sandwich if necessary.
If you do find you get a partial cut, you might still be able to use the shape if it's a reasonably simple one - just pop the die out and use the groove in the board to guide sharp scissors around to finish the cut.
Steel rule dies (like the Bigz ones) will cut through really thick materials (mountboard, Grungeboard etc) so if you have any of those they'll certainly cut through the board you have.