I'm going to take a guess and say that your rings are mis-aligned when you're trying to clamp the rivet. Open the rings "up and down" as carefully as you can, and when you have it woven, use the pliers to "close" the ring overlap before setting the rivet. If the ring is putting tension on the rivet, it can set crooked like that when squeezed; my first couple did. The rivets are TINY so it takes almost immeasurably small amounts of pressure to tweak them out of alignment.
If that's not the case, open the pliers, close them slowly, and make sure the "dimples" for the rivet line up. If they don't, the pliers are out of alignment (and easy to adjust). If they do line up, just work on closing the links more before riveting - this would be my guess.
My hat's off to Allan for setting them by hand; I'd be driven to the pits of madness and despair trying to pein those tiny rivets!