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.