Why are volume and pressure inversely proportional to each other? It makes sense, that if you have a balloon and press it down with your hands, the volume will decrease and the pressure will increase. This confirms Boyle's Law, $ pV=k=nRT $.
But what if the pressure in the balloon increases? Doesn't it make sense that the balloon would want to expand? That is, that as pressure increases, volume increases. This seems to contradict Boyle's Law.
Could you explain what I'm doing wrong in the second scenario?
Could you give an example that does confirm Boyle's law?
 A: Pressure and volume have an inverse relationship when $n$ and $T$ are constant. How do you imagine the pressure in the balloon is increased? Either $n$ or $T$ must increase, or $V$ must decrease.
Additionally, balloons are roughly constant-pressure systems. The rubber membrane is a very weak elastic, so the internal pressure of the balloon is at almost constant pressure, just above atmospheric. When you squeeze a balloon, you usually don't change the pressure or volume much, because the rubber just expand in an area where you aren't pressing.
Since you edited your question to ask for an example of Boyle's Law: Consider a piston in a cylinder. As the piston is pushed in, the gas in the cylinder is forced in to a smaller volume, and its pressure increases. If this is done slowly enough that no significant heating of the gas occurs, the relationship of pressure and volume will follow Boyle's Law.
A: Think about this: why is the pressure increasing? If it's because you're blowing air into the balloon (which is the usual way to increase the air pressure), then what you're actually doing is raising $n$. And it makes sense that an increase in $n$ should be correlated with an increase in $p$ (or $V$, or both). Boyle's law doesn't apply in this case because $n$ isn't constant.
In fact, for a balloon, as you blow air in to increase $n$, you wind up increasing both $p$ and $V$. The reason for this is that the balloon is elastic: as you blow it up, it "wants" to return to its original, uninflated size. It basically acts like a spring. As you put more air in, the balloon has to get bigger to contain it, but also the bigger it gets, the harder it pushes to try to return to its original size. So it squeezes the air inside it a little more, and the pressure goes up.
A: The problem with gasses is that you have also a third parameter - temperature, and this third parameter decides in which direction is balloon under pressure change going to go.  In first case - increasing pressure and decreasing volume, you keep your balloon at constant temperature.  In other case you have to let temperature to rise.
How can you control in which direction will balloon go?  Well, you can keep temperature constant by having good thermal connection to the environment with a constant temperature.
Example of a concomitant rise in volume and pressure is engine, i.e. piston in the cylinder.  What happens there is that both pressure and volume of the gas within cylinder increase and the increase in volume keeps engine running.  But to force gas to increase both volume and pressure, you must massively increase its temperature.  You do that by making an explosion within the cylinder.  (Actually all explosions are concomitant increase in pressure and volume and all are based on huge increase of temperature!)
So the temperature is the key.
A: 
But what if the pressure in the balloon increases? Doesn't it make sense that the balloon would want to expand? That is, that as pressure increases, volume increases. This seems to contradict Boyle's Law.

In simple words: If you increase the pressure in the balloon and let it expand, then the pressure in the balloon is not really increasing, as you are also letting the volume increase and thus overall your $k$ remains constant. In order to actually increase the pressure means that you have to stop the expansion in volume, only then will you have more pressure. If you let volume go up then you are letting the pressure stay constant even if you are feeding more air into the balloon. Only way to increase the pressure is to let the volume stay constant, and if you stop the expansion of the balloon somehow, then you will indeed have an increased pressure inside it on feeding more air into it, and Boyle's law will remain intact.
(assuming a balloon in which the expansion is perfectly proportional to increase in pressure)
A: Assuming you have an perfectly elastic balloon and an ideal gas, AND keeping all other variables constant (e.g. n and T), then no. If you had some magic button to increase the pressure, then the volume should decrease. It's simply the mirror image of your predicate: "if you have a balloon and press it down with your hands, the volume will decrease and the pressure will increase." For example, "If you have a balloon and press the increase pressure button, the volume will decrease and the pressure will increase.".
To answer your original question, "why are pressure and volume inversely proportionaly? You need to know a little about Statistical Mechanics and and Calculus. But basically, Pressure is only a mathematical concept. But Volume and Energy are measurable (extensive) quantities. Pressure is defined to be proportional to the rate of change of internal energy relative to volume. Just like velocity is a mathematical concept defined to be Distance/Time. Velocity is inversely proportional to time in the equation V = D/T. Just like the relationship between pressure and volume.
Velocity or Pressure can not be measured or observed directly. Only by measuring at least two extensive properties (distance and time, or energy and volume) can you define a value for either. Your confusing likely is grounded in the difference between intensive and extensive properties.
A: The two scenarios you mentioned both are correct, “the pressure $p$ has different sign from other generalized force, if we increase the pressure, the volume increases, whereas if we increase the force, $Y$, for all other cases, the extensive variable, $x$, decreases”.[1] There is no conflict between the two scenarios.
[1] L.E.Reichl, A Modern Course in Statistical Physics, 2nd ed.(1997), p23
