People just need to lie more often. In many cases, they will not verify anything. If they require proof, you can say everything was a verbal offer with no official documentation. That's often how it is because many recruiters don't want to go through the paperwork process before there's an agreement.

That said, I've found the "competing offer" element to be of little to no value. Even when I had 7 competing offers, not one of them negotiated more than maybe 10% on the amount. The one that gave me the largest bump didn't even need to know about competing offers - they just wanted to know what would get me to sign. I said X and they did it.