Question: If I feed a duplex receptacle from two separate branch circuits, I'm supposed to remove the jumper tabs between the receptacles. But what would happen if I didn't remove the tabs?

Answer: 
It's worse than tester101 says. Each circuit may not see half, there may be an imbalance of currents between the redundant paths. (Especially if one has a problem, such as being completely broken). Now how are the wires protected from overload? The hots have rather nice breakers on them, but the neutrals are not breaker protected! Nothing will detect an overload there. 

[![enter image description here](https://i.stack.imgur.com/YXHzz.png)](https://i.stack.imgur.com/YXHzz.png)

The opposite is an even bigger problem - suppose current flow does balance. One circuit has *other* outlets, and those are overloaded to 27A. They draw through this outlet, drawing a relatively balanced 14A from one breaker and 13A from the other. Whoops! 

[![enter image description here](https://i.stack.imgur.com/ESpL2.png)](https://i.stack.imgur.com/ESpL2.png)
If you add two live 15 amp lines to one outlet you will blow the breakers if not the main.