Why is it said that antiparticles are a result of combining SR with Quantum theory? I did understand the historical reasons for the discovery of antiparticles in this context. But are antiparticles really a 'consequence' of combining special relativity and quantum theory? Why isn't it better to say that the existence of antiparticles are consistent with QM and SR?
 A: Here is another explanation, due to Richard Feynman and outlined in his Dirac Lecture (of which a video is actually available on youtube):
He starts with a space-time diagram, as used to track interactions and events in special relativity, and the equations to calculate probability amplitudes in quantum mechanics. He then sets up a problem in the diagram of a particle scattering off a potential, moving for a bit, and then a little while later scattering off a second potential which returns the particle to its exact same initial state.
He then calculates the probability amplitude for the double-scattering problem using QM and finds that for the probability to be nonzero, the second scattering must be located outside the light cone of the first scattering (this is magic).
This means that it is possible for an observer in a certain state of motion relative to the two scatterings to observe the second scattering before the first one has happened, and to conclude that between the two scattering events, the particle was traveling backwards in time. He supports this by drawing the two-scattering event from that frame of reference, which shows the world line of the particle zig-zag backwards between the two scatterings.
He then assigns an arbitrary electric charge to the particle and asserts that since particles can't travel backwards in time, the physical situation is indistinguishable from one in which an oppositely charged particle is traveling forwards in time.
The spacetime diagram that results has a particle moving forwards in time, minding its own business, while a short distance away a particle/antiparticle pair bursts into existence. The antiparticle then hits the original particle, annihilating it, and leaving the particle half of the pair motoring along in place of the original particle.
He then points out that even though there appeared to be three particles in play during the time slice between the two scatterings, there really was only one particle all along, whose world line was briefly kinked backwards in time before continuing on its merry way.
Hence therefore, Feynman concludes that putting relativity together with QM yields antiparticles.
A: There are perfectly well-defined non-relativistic theories that contain anti-particles, and perfectly well-defined relativistic theories that contain no anti-particles.
Examples are a dime a dozen. Take a complex scalar field and turn on a Lorentz-violating interaction. Poof: now you have a non-relativistic theory that contains anti-particles. Conversely, take a scalar field and do Lorentz-invariant interactions only. Naturally, everything is its own anti-particle, and the system enjoys full relativistic invariance.
In conclusion, there is no connection between special relativity and anti-particles.
Not even field-theory is related to anti-particles, at least to the extent that you can define a satisfactory notion of "particle" for non-field theories. You can do e.g. a quantum mechanical model a la SKY with complex fermions. This is not a field theory, yet there is a perfectly sensible notion of anti-particle, as inherited from the charge-conjugation symmetry.
A: This is a very interesting question. Galileo once stated that "our Universe is a “grand book” written in the language of mathematics".

But are antiparticles really a 'consequence' of combining special relativity and quantum theory

Yes, they are, in that the relativistic mathematical formulation of quantum theory suggests that antiparticles emerge from this mathematics.
In going from standard quantum mechanics to quantum field theory, we introduce the Klein-Gordon equation $$\pm \sqrt{m^2c^4+{\vec{p}}^2c^2}\; \psi = i \hbar \frac{\partial}{\partial t} \psi$$ and the Dirac equation $$(\beta m c^2 + c {\vec{\alpha}}\cdot {\vec{p}})\Psi = i \hbar \frac{\partial}{\partial t} \Psi$$
These equations have solutions that represent both positive and negative energies. We are familiar with positive energies, but this appearance of negative energy solutions represented a possible conundrum, and so a closer look at was happening was required.
Negative energies were nonsensical, and to make a long story short, these negative energy solutions are explained by the existence of antiparticles.

Why isn't it better to say that the existence of antiparticles are consistent with QM and SR?

It's OK either way. The existence of antiparticles is consistent with the combination of QM and SR, and the mathematical formulation itself also predicts the existence of antiparticles.
A: Due to QM, you must have wave equations.
Due to SR, you must have relativistic dispersion for your waves. That means,
$$ E^2 - p^2 = m^2 $$
with $E = \omega$ the energy/frequency of the particle/wave (in natural units, this is the same quantity), $p$ the momentum / inverse wavelength, and $m$ the mass of the particle/field.
Unlike its nonrelativistic approximation
$$E = m + \frac{p^2}{2 m}$$
(I've taken the first two terms in the series corresponding to rest energy and the first kinetic energy term) that leads to Schrodinger equation (with $m$ an unobservable shift in the particle's phase that is usually subtracted from the Hamiltonian for convenience), the full relativistic dispersion relation is quadratic in $E$, not linear.
That means that for any theory that is both Quantum Mechanical and Relativistic, we always have negative-energy solutions that correspond to positive-energy solutions.
This is probably a better statement from the mathematical point of view. QM and SR combined give rise to negative energy solutions, not necessarily anti-particles.
To actually make sense of these negative energy solutions, you need the second-quantized formalism (which is closely related to the first-quantized formalism, essentially you just take the Fock space built on top of the first-quantized space of states). Then your solutions can be interpreted as (superpositions of) creation / annihilation operators for two sorts of particles that must have identical properties except for charges which must be conjugate. That's your antiparticles.
I have no comment about anna's answer or "platonists" mentioned therein, I don't believe math molds nature or the other way around, I believe math is a way to model nature. This answer explains why it is usually assumed that QM and SR impose enough constrains on the types of models that you can built that you always should expect negative-energy solutions. One meaningful way of making sense of those is to introduce second quantization and antiparticles.
A: It is a shorthand for "in order to mathematically model successfully the experimental data and the  existence of antiparticles, one needs both special relativity and quantum field theory".
Physisista who truly see it as a result belong to the class of "platonists" believing that mathematics molds reality, not that reality is modeled using mathematics.
Edit after a number of negative votes :
The historical fact is that if there had been no observations of particles with the exact same mass and opposite quantum numbers, there would be no incentive to model antiparticles, and any such appearing in mathematical models would have been sent to huge mass values, the way we do with super-symmetric  particles etc now. Data drives mathematical models.
Edit 2.
It is observed that the Dirac equation predicted the existence of antiparticles.  It is the correct fits  for existing data that validated the Dirac equation, and the theory predicted the antiparticles, but the antiparticles are not the "result" of the theory. They are the result of the theory fitting well and predicting the particle data of that time, so that its further predictions could be "trusted".
In my answers I try to ground models to observations, which is what physics is.
