The title and the cartoon itself are parodies of The Defiant Ones, a movie about two escaped inmates shackled together.
Sylvester and the unnamed bulldog (both voiced by Mel Blanc) are on a truck headed for the pound. Both are shackled to each other and hate each other's guts. When the truck hits a bumpy road, the truck's back door opens, dumping Sylvester and the bulldog out. The two quickly run for cover, which is easier said than done due to being stuck together.
After fighting over some food, Sylvester and the bulldog realize they must break the chains. However, attempting to blow up the chains with TNT and hammering it both fail. Later, while walking near railroad tracks, a train snags their chain. That night, the two sneak into the city and take brief refuge in an abandoned building when a spotlight almost catches them. They decide to disguise themselves in human clothes (with the other person hiding in a suitcase) and hop aboard a bus, but hightail it out of there when the bus is revealed to be headed for Sing Sing.
In the final set piece of the cartoon, the bulldog gets an idea to hang below a train track that goes between two mountains, with the idea that the train will cut their chains. However, both realize it's a bad idea too late when Sylvester points downward, indicating that once the chains are broken, they'll fall hundreds of feet to the ground. Sure enough, Sylvester and the bulldog fall into a junkyard after their chains are cut. At first, both continue to argue about how dumb the plan was, but are briefly joyous when they realize they aren't shackled together anymore. However, they look down and realize they're now connected by leg via a pipe. They hear police sirens and hop down the road to escape as the short ends.
On the ABC version of this cartoon, the following scenes were cut:
Hector punching Sylvester in the face after Sylvester pulls "a switchblade claw" on him.
Hector whacking Sylvester on the head with a stop sign.
The scene where Sylvester and Hector try to escape by hiding each other in suitcases, only to end up at a bus stop for buses heading to Sing-Sing Prison and the two running off when they discover the dreaded destination.