If you mean blow it to pieces as in Star Wars breaking up in seconds with the pieces flying away at way over escape velocity - no, nowhere near. But that is a totally silly way to blow up a planet :).

See this video for a calculation of the energy required and the physics of blowing up a planet Star Wars style.

If you mean a large asteroid impact like the one that ended the dinosaur era - yes probably, though not easily. Yes there are lots of 10 km scale Near Earth Asteroids - nearly 1000 of them in total. Any of those could in principle be deflected to hit Earth, but it would take a while. Unless one of them happens to be headed our way.

When they talk about the possibility of deflecting these asteroids - it's a case of changing it's path by the radius of the Earth over, say a decade or two. Maybe even more quickly than that with nuclear weapons.

Could be just by a few  hundred meters though if they do a flyby of Earth in between and have to go through a crucial "keyhole" in order to hit Earth next time around.

That's not much considering that they typically miss Earth by more than the distance to the Moon. The distance to the Moon is 30 Earth diameters, or six Earth radii.

But an impact like that is one that many humans could survive, especially as we'd see it easily years in advance, as we track all those NEOs - we currently know all the ones of 10 km or larger. Even if our opponents are able to stop us from doing anything about it, e.g. military extremist future space colonists who want to destroy the Earth, nightmare future scenario - able to destroy our rockets as they leave the Earth - well still if you get into a submarine, or use fire shelters, stock up on food, maybe some oxygen just for the moment of impact, many would survive.

Indeed, at the moment of impact probably many would survive anywhere a few hundred kilometers outside of the impact zone, though you'd then need to watch out for large meteorite debris from the sky, and firestorms.

After all birds, mammals and turtles to name a few species all survived the dinosaur extinction, and where they can go to survive, we can also with our technology.

If you want to melt the entire crust of the Earth and evaporate its ocean, as happened with the impact that created the Moon, you need a larger impactor. At least a hundred kilometers in diameter. There are plenty of asteroids that big, but it would take much longer to deflect them. Probably best way is to do "asteroid billiards" deflect smaller asteroids, which is far easier to do - and use those to deflect larger ones, and those to deflect even larger ones. The idea is that there will probably be some smaller asteroids that happen to be going in the right direction to give the bigger one a big wallop if they happen to hit it - but always miss. So you deflect them just enough to hit. And those then you deflect just a bit to hit larger asteroids and so on - in a carefully co-ordinated campaign like that, stretching probably over a centuries or millennia, it might be possible with our present day technology - though of course technology would also advance in the meantime.

For the Moon itself, it was a Mars sized impactor, and even that didn't blow the earth into pieces as in Star Wars:

None of this is going to happen naturally though.

We'd see it coming centuries in advance, and it is so improbable, that we can forget about it. The impact history of the inner solar system shows that there have been no large impacts by asteroids larger than 10-20 km in diameter on Mars, Earth, Moon, Mars' moons, Mercury, and what we have of the cratering history of Venus since it's global resurfacing a few hundred million years ago.

And hopefully we are not headed for a future with militaristic space colonists. If we are, well they would be far more fragile than any Earth colony. So unless they had a total monopoly on space technology somehow, can't see them winning. They'd soon destroy themselves or be destroyed in their fragile habitats with nowhere outside of Earth where they can breathe without a spacesuit. And unless deeply buried, any space settlements would be very vulnerable to the kilometers per second impacts you can achieve easily once you have space technology.

Not too likely that they are able to go through the entire process of deflecting asteroids over many centuries with humans on Earth unable to do anything about it. I think the space colonies would be the first to go in any interplanetary warfare.

However, we have luckily managed to avoid warfare in space totally. The outer space treaty ensured that. Before then the US were planning a military base on the Moon. But they dropped the plans when they signed the treaty which prohibits any nation state from claiming territory in space, or siting military bases on celestial bodies, or putting weapons of mass destruction in space. Using an asteroid to destroy the Earth obviously would be a huge violation of the space treaty :).

It's worked so far, and it's in the interest of everyone I think, to make sure that we continue with peaceful exploration of space. Everyone can see that. Even N. Korea has both signed and ratified the Outer Space Treaty.