Do I have grounds to sue the dealership for the damage done to my car?

My car jumped time so I took it to the dealership. The replaced belt told me the camshaft was bent, but the car was running, replaced it but during the replacement process a piece was broken off and fell down inside motor. They installed a new camshaft, turned key, and locked up the motor, sent it to machine shop for repair, but was too far gone to fix, I got to have a new motor, now they say it wasn't them, that it was already like that before they changed camshaft, but it wasn't because it would not have been running. They had the car for 3 weeks and 2 days before the motor locked up, and it's been an additional 2 weeks and 4 days, but still no replacement engine found, and now they say they won't make it right unless I pay for everything.