Isn't the uncertainty principle just non-fundamental limitations in our current technology that could be removed in a more advanced civilization?

Manishearth's answer is correct, and this is just a minor extension of it. Manishearth correctly points out that the problem is your statement:

There is a definine velocity and momentum, we just don't know it.

Your statement is the hidden variables idea, and courtesy of Bell's theorem we currently believe that hidden variables are impossible.

Take the example of a hydrogen atom, and ask what the position of the electron is. The problem is that properties like position are properties of particles. It doesn't make sense to ask what the position is unless there is a particle at that position. But the electron is not a particle. The question of what an electron really is may entertain philosophers, but for our purposes it's an excitation in a quantum field and as such doesn't have a position. If you interact with the electron, e.g. by firing another particle at it, you will find that the interaction between the particle and electron happens at a well defined position. We tend to think of that as the position of the electron, but really it isn't: it's the position of the interaction.

The uncertainty principle applies because it's not possible for an interaction, like our example of a colliding particle, to simultaneous measure the position and momentum exactly. So you're sort of correct when you say it's an observational limit, but it's a fundamental one.

There is a definine velocity and momentum, we just don't know it.

Nope. There is no definite velocity--this was the older interpretation. The particle has all (possible) velocities at once;it is in a wavefunction, a superposition of all of these states. This can actually be verified by stuff like the double-slit experiment with one photon--we cannot explain single-photon-fringes unless we accept the fact that the photon is in "both slits at once".

So, it's not a knowledge limit. The particle really has no definite position/whatever.

Isn't that equivalent to saying because we haven't seen Star X, it doesn't exist? It's limiting the definition of the universe to the limits of our observation!

No, it's equivalent to saying "because we haven't gotten any evidence of Star X, it may or may not exist --it's existence is not definite" Technically, an undetected object does exist as a wavefunction. Though it gets slightly philosophical and boils down to "If a tree falls in a forest and no one is around to hear it, does it make a sound?"

This seeming leap is an invocation of logical positivism. Logical positivism is the default philosophy in physics, it is indispensible, and it has been the source of nontrivial ideas which have been crucial to progress for over a century.

You can't assume that there is a position and momentum simultaneously in the particle, because this point of view would lead you to believe that there is a probability for the position and the momentum, and that each possible position and momentum evolves independently. This is incompatible with observations. No independent position, momentum picture can be different from Newtonian, classical mechanics.

You can see this, because a wavepacket with a nearly definite momentum moves as the classical particle, a wavepacket with a nearly definite position is at a spot, like a classical particle, so together, if both are well defined at all times, the particle would be moving from definite position to definite position as in classical mechanics. This is impossible, because it would lead to sharp trajectories and no diffraction of electrons around objects. Electron diffraction is observed every day.

One can still claim that the position is a hidden variable, and not the momentum, but then the momentum is only partly defined, as a property of the carrier wave. This is what happens in the Bohm theory.

The reason one cannot assign hidden variables in an obviou way to the particles in quantum mechanics is because the calculus regarding the different possibilities is not a probability calculus, but a calculus of probability amplitudes, and probability amplitudes don't have an ignorance interpretation.

To see this, consider a particle which can goes from a state $|0\rangle$ (where some physical bit describing its position is 0) to the superposition state $|0\rangle+|1\rangle$ and from $|1\rangle$ to $-|0\rangle + |1\rangle$ in a certain period of time, say 1 second. Now start off in the state ($|0\rangle+|1\rangle$), what happens? By linearity, you end up in the definite state $|1\rangle$. So if you consider "1" a definite state, it becomes more uncertain, but in the uncertain combination, it recongeals to become certain! This doesn't happen in probability, because different probabilistic branches can't combine with a minus sign the way that they just did above in quantum mechanics to get rid of the $|0\rangle$ component.

The sign issue makes the ignorance interpretation of quantum mechanics untenable--- only probabilities are ignorance, and only in the limit of very large systems does quantum mechanics (approximately) reproduce something like probability. This involves observation.

The way in which the theory was constructed was by carefully applying logical positivism at each stage, and if one does not internalize and apply positivism, you don't get the theory. See this related answer: How can indeterminacy in quantum mechanics be derived from lack of ability to observe a cause?