How can a horse move a cart if they exert equal and opposite forces on each other according to Newton's third law? Imagine a horse is tethered to a cart. According to Newton's third law, when the horse pulls on the cart, the cart will also pull backwards on the horse. Since the two objects are attached together, they are technically the same object, and they cannot accelerate.
This doesn't make any sense. In the real world, horse carts are able to move, even though they are attached together.
Am I overlooking something here?
 A: You are overlooking the force between the horse and the ground.
Yes, if you had only the horse and the cart with nothing else, then the system could not accelerate as a whole (i.e., the center of mass could not accelerate). However, friction between the horse and the ground pushes on the horse, thus allowing for an acceleration of the system.
A: You are forgetting that there are other forces acting on the horse and cart as well as the force that the horse exerts on the cart and the equal and opposite force that the cart exerts on the horse (and innumerable other pairs of equal and opposite forces between neighbouring parts of the horse-cart system). The most important of these other forces is the frictional force that the ground exerts on the horse's hooves. It is this forward frictional force (assuming that the horse is co-operative and pushes backwards on the ground) that provides the acceleration of the horse and cart. If the horse and cart were on a sheet of wet ice, it wouldn't happen!
A: 
Since the two objects are attached together, they are technically the
same object

So what? The horse-cart combo is applying a force to the ground via the horse's hooves. The ground provides an equal and opposite reaction to the hooves so either the horse-cart will move forward or the Earth will spin in the opposite direction. In fact both happen but the spin imparted to the Earth by the horse-cart is  so tiny as to be negligible. Therefore we say that it's the horse-cart that moves.
Notes

*

*The tether between horse and cart merely transfers the force (of the hooves) from one to the other. It does not make them accelerate relative to one another - they both accelerate relative to the ground.


*If the friction provided by the cart against the ground prevents the combo from moving (say the brakes are on) then nothing accelerates relative  to anything. In that case your proposed scenario is true.
A: 
Since the two objects are attached together, they are technically the same object, and they cannot accelerate.

That's not really how it works here.  That's like saying since a car is the same object, it cannot accelerate itself relative to the ground.
If the horse were rigidly attached to the cart (as opposed to "tethered", which implies a rope), it would struggle to accelerate relative to the cart, but both the cart and the horse can be accelerated relative to the ground by applying force on the ground (when the horse runs).
A: You are correct in suggesting that the cart pulls back on the horse. The best way to picture this is to consider a horse attached to a cart by a 100m rope coiled up. The horse is allowed to run away as fast as it can. When the rope runs out, the horse will most likely get pulled almost to a stop in what will be a very dramatic surprise for the horse! But the cart will start moving. However much momentum the horse had when it hits the end of the rope will be transferred to the cart. So the cart does indeed pull on the horse, but in doing so receives some (probably most) of the horse's momentum. That's how Newtonian Physics works. Newton's Laws basically evaluate to the following statement: Momentum is preserved in any system that receives a zero outside force, and the change in momentum of any system is the change in the product of the mass and (directional) velocity of the system. If we can assume that the horse and the cart represent a closed system, then if the horse comes down to 10% of its original speed, the velocity of the cart will be the horse's decrease in velocity multiplied by the ratio of the horse's mass to the cart's mass. So if the cart has 2x the mass of the horse, it will be moving at 45% of the horse's speed and in the same direction because the horse pulled on the rope, the rope pulled on the cart, accelerating it, the cart pulled back on the rope, and the rope pulled back on the horse, stopping it. Now if the horse burns some of the grass it ate and uses it to generate some energy which it then transferred to the earth using its legs, we would need to include the Earth in the momentum system. Since momentum has to be preserved, however much momentum the horse imparts to the Earth, the Earth will push back. The horse will appear to be moving relative to the Earth, but keep in mind that the Earth has been pushed in the opposite direction. Now bring the rope's length down to essentially zero and consider this chain of events being repeated rapidly. The reason why your original point doesn't work is because in some cases the system consists only of the horse and the cart, and sometimes it consists of the horse and the Earth. First the horse does a momentum exchange with the Earth, then it does a momentum exchange with the cart. This process repeats over and over again and the (horse + cart) system is moving relative to the earth.
A: The horse is harnessed or yoked to the cart, not tethered.
The overall assembly, i.e. comprising "horse" and "cart" with an arbitrarily amount of compliance in the coupling, includes 4x subassys designated as "legs with attached hooves". These operate in such a way as to impart force on the surface on which the horse and card are placed, with the result that the horse+cart assy move in one direction and the surface moves in the other direction, resulting in relative acceleration proportional to their masses (Newton's third law).
Since even the largest horse and cart weigh substantially less that the Earth, they will move forwards.
A: 
Imagine a horse is tethered to a cart. According to Newton's third law, when the horse pulls on the cart, the cart will also pull backwards on the horse. Since the two objects are attached together, they are technically the same object, and they cannot accelerate

Let's change the situation a bit, and change the setting to - space. Replace the horse with an astronaut, and replace the cart with something heavy - maybe a toolbox of some sort. The astronaut is freefloating outside of the International Space Station, doing some maintenance work. When the astronaut pulls on the toolbox (via the tether), the toolbox will also pull on the astronaut. These two forces act on separate objects, and are directed at one another.
The astronaut and the toolbox collide.
So it's not that they cannot accelerate - these two forces don't cancel out, as they aren't acting on the same object. (Or, if you want to treat them as a composite object, keep in mind that it's not a rigid body.)
Well, the same thing happens with the horse/cart scenario, except that the horse has the ability to resist the reaction force because it is in contact with the ground. The cart itself rolls on wheels and will start to go when pulled - if we ignore friction, and vertical forces (gravity, normal force), there's a single, non-zero force acting on it. So why the cart moves is not mysterious. Of course, the horse moves as well, and the two act roughly as a unit, so we can say that both the horse and the cart have the same acceleration. For this to be true, the horse needs to produce a forward-directed force, larger in intensity than the reaction force, so that the net force on the horse provides that particular acceleration (the image below I found here shows a similar situation; it ignores friction, drag, etc.).

One point that's perhaps counter-intuitive is the fact that the forward-directed tension force doesn't have the same magnitude as the propulsion force $\vec{D}$ produced by the horse/truck.
If $\vec{F}$ is the net force on the horse/truck, note that $D = F + T = m_1a + m_2a = (m_1 + m_2)a$, so if you consider the system as a whole, $\vec{D}$ acts on the combined masses of the two objects, while internally the tension $\vec{T}$ only acts on $m_2$. (Note also that in this model the objects are treated as if connected by an idealized massless cord.)
A: 
Since the two objects are attached together, they are technically the same object, and they cannot accelerate.

This can be applied to all objects. All objects are attached together by some force (be it gravity, the electromagnetic force, the color force, or the weak force).
But the cart and the horse are normally seen as two separate objects. The horse pulls on the cart, the cart pulls on the horse (action=reaction).
So if the horse pulls on the cart, say not by means of friction with the ground it walks on but by means of a rocket flying through space [like Santa Clause' reindeers pull Santa's cart (or do the run in the air?)], the cart gets accelerated. The cart pulls back (like the whole system of the cart, horse, and rocket wrt to the propulsion jet coming out of the rocket). This is because the cart, as well as the whole system, has inertial mass (and every sub-system has a correlated sub-inertial mass).
A: You forget that horse also applies force on the ground so that it can move together with cart. The forces between horse and cart only keep them relatively steady with respect to each other.
