They can pretty much do whatever they like. A lot depends on your attitude. There's a pretty extensive "How to get out of tickets" text file floating around the internet that was written by a former state trooper, and he basically says that it all comes down to a) your attitude to the cop, and b) how reckless your speeding was.
That said, I'm always damn near obsequent to cops, and the only time I haven't gotten a ticket is when the cop asked me how fast I was going, and I truthfully said "71 or 72", even though it was in a 55 zone, the cop thought my out-of-state license meant that I didn't know the limit had dropped from 65 in that section of interstate. I did.