http://www.klickitat-trail.org/images/goathead.jpg
I've gotten a good number of flats from these, a broken point can be hidden within the tire. I rub talcum powder on the inside of my tires, I often feel something hiding there.
Pay attention when changing your next flat to the tube and tire relationship it will narrow the search if the problem is the tire or rim.
As others have mentioned the likely culprit is not enough pressure.