I have an ever better question... WHY would you even want to try that? What was you hoping to achieve?

Sure you can. It is a very common event.

Its called a "Split Receptacle" Normally the top of the recptacle is a constant hot and the bottom is switch controlled. This is one way to use floor lamps and turn them on with a switch.

A multi-wire circuit is a perfect feed for a split receptacle.

Donf is correct! Just make sure you pull the metal bonding tabs on each side of the receptacle before using it in this fashion.

For the application Donf mentions, don't remove the neutral tab.
I have done small Kitchens where all were split recepticles(2 circuits). Can't trip a breaker.

In completely electrically independent split wired receptacles, each receptacle also has its own independent neutral wire and possibly ground wire back to the electric panel. In a multi-wired or shared neutral receptacle, a single neutral wire is shared by both of the independently powered receptacles. In the case of using a common neutral wire or having the receptacle neutral bonded you may exceed the rating of the neutral wire if both circuits were nearing their maximum allowed current individually, and if on the same bus in the main panel would result in both circuits being in the same phase. This could also cause the neutral wire on one circuit to be energized while the other was thrown at the breaker. Someone working on a circuit that was intentionally thrown at the breaker could be shocked if the other circuit was being heavily used and that person touched the neutral and ground wire at the same time. Best practice is to isolate each circuit in the split receptacle to avoid these problems. If you decide to use a common neutral it is recommended to use a double pole breaker for the two circuits to keep them on separate busses and out of phase with each other and to disable both circuits if one were to have the need for maintenance or modifications. I am still at a loss as to why the split receptacles wouldn't throw a breaker though, even if they were using the same neutral. Both methods should work but one has inherent dangers... Kind of hard to explain but hopefully someone will get something from it.

Any time two circuits supply a single device, such as two circuits connected to a duplex receptacle, no matter one or two neutrals, a 2 pole circuit breaker must be used.

Home inspector, quote me where I was wrong.

Why would someone not pull the neutral tab and add an extra level of protection when installing a multi-wired split receptacle? There is absolutely no benefit, that I can think of, to not pulling the neutral tab and potentially creating a hazard if you don't, especially if they weren't properly wired on a double pole breaker, which I'm sure many are not. It would/could cause issues, not only with the neutral wire being undersized to carry 2 separate loads but with the added risk of electrocution for anyone working on one of the two circuits and thought they had it terminated by throwing a breaker. You could very easily reduce these risks by simply removing the neutral tab. Telling someone they shouldn't remove the neutral tab is not sound advice because of that. In my opinion

There are two situations with having two circuits feeding one device.

First, which is typical, using a three wire cable, one shared neutral for both circuits.

In this case, the neutral jumper must not be removed.

If two circuits serve one device, and each circuit has it's own neutral, only then shall the neutral jumper must be removed, so each of the two receptacles on one duplex device can have its own live and neutral.

I don't understand how the other receptacle will work with the TAB off.
Wiring correctly is not to difficult to place on a 2 pole Breaker/GFI/ARC Fault Breaker.
They will automatically be on opposite phases.
1 Neutral for 2 circuits is very common and is usually most dangerous when the neutral is open with power applied, 1 voltage can go way high and the other way low. Both Dangerous.

Ah I see, I wasn't considering a 3 wire circuit. Was only thinking about 2-2 wire circuits. I guess using a 3 wire circuit wouldn't allow for a second neutral wire and if that is what you were referring to Strat my apologies. I guess the multiwired part of the question had me thinking more than one cable feeding the receptacle. I should have thought about that as I have seen kitchens wired with 3 wire and each receptacle alternated hot wires to reduce the draw on each wire. Wouldn't this still potentially overload the neutral wire if both hot feeds were nearing their maximum potential?

If on opposite phases, if 1 circuit draws 20 amps and the other draws 15 amps, the neutral would draw the difference of the 2, which would be 20-15= 5 amps.
If both were on the same phase, the Neutral would draw the total of the 2, 20+15=35 amps and the Neutral would be overloaded.

It should also be pointed out that tie bars on 2 breakers nor tandem breakers in this circumstance are not allowed. I think this is one of the changes NEC has made since the last publication, I think tie bars used to be permitted. Still not to sure about the original question referring to "different breaker feeds being wired into one receptacle" was speaking of a 3 wire or 2-2 wire circuits. I guess we should have clarified that before proceeding with an answer, but in either case the answer would be yes...