Question: In my story, I want a creature to be immune to bullets, but giving it bulletproof skin seems too obvious. I want it so that you could shoot the creature a few times at its seemingly vital spots (the head, heart, etc.), but it won't die right away and can still escape or attack you. The best way to make sure it dies quickly is by draining most of its blood, which means you have to get close enough to cause a large wound with a bladed weapon.

Yes, I know you can just use a really big gun or explosives to achieve the same effect, but in this scenario, the options are limited to something relatively cheap, lightweight, and doesn't cause a lot of collateral damage. To give you some image, the creature is about as agile, as strong, and as big as a polar bear.

Preferably the reason it's immune to bullets is based on real animals, or at least something that could exist in the natural world.

Answer: 
What about a dense skeleton?

The skull would protect the brain of course, and you could have a ribcage where the ribs are more tightly packed (or even overlapping dual layer, allowing expansion in both layers but still protecting against vital shots) so that lungs, heart et al are protected. You still need a circulatory system so the idea of large trauma from a blade still works.

Sure, you could get a shot into the arms or legs, but that would not be fatal. Even a stomach wound wouldn't kill the animal immediately meaning that it could still attack you.

It violates the possibility of the brain, heart or lungs being shot, but still provides for firearms to do damage generally, just not in critical areas that would cause immediate death.

This does have some precedent in nature; certain herd animals have hardened skulls, and there are plenty of dinosaurs with hardened skeletons as they used parts of their bodies as clubs, rams or spears.
Redundancy.

Bigger than necessity organs due to evolutionary pressure to survive wounds and duplicated vascularisation with the possibility of closing many veins/arteries if severed.

Wherever you hit, there will be another part of the organ able to take over the function and the blood drain will be minor because the tissue can survive a long time without blood while healing and the arteries are closed or almost closed.

People can live with [literally half their brain missing](http://www.bbc.com/future/story/20141216-can-you-live-with-half-a-brain), make that per design and you can destroy half the head without problems.

Bonus: to kill it you need to severe both vital arteries at one of the few points where they are near each other.