@Uppi - why are you skeptical of supersymmetry? I thought that the observation of anti particles pretty much confirmed a lot of it.
Are there specific predictions that haven't worked out even though the experiments were well-designed?
No, supersymmetry and antiparticles are not really related. We have known since the thirties (positron) that particles (can) have anti-particles. An anti-particle will have the same properties as the particle, but sometimes with a different sign (let's ignore CP violation for a moment). E.g. a positron has the same mass as the electron, and the same charge, but with an opposite sign. Importantly, the positron also has the same spin (1/2) as the electron. This makes the positron and the electron so much the same that they are actually described by a single field in Quantum Field Theory, and one generally considers them to be just different manifestations of the same particle. The anti-particles of other elementary particles usually don't even have their own name, and for some of them the distinction between particle and antiparticle is rather arbitrary. (You will also sometimes hear people say that an anti-particle is just a particle travelling backwards in time. This is a way you can interpret anti-particles, but it is little more than an interpretation.)
Now, as you might know, there are two types of particles in nature. Particles with integer (0,1,2...) spin, which are called bosons and particles with half-integer (1/2, 3/2,...) spin, called fermions. These types of particles behave radically different. Supersymmetry predicts that every fermion has a supersymmetric bosonic partner and vice versa, but that these partners just have not been discovered yet.
Such a theory is somewhat appealing, since it unifies our view of nation even more (and unification is good!) and I believe it also might explain some aspects of the Higgsmechanism. However, the only real prediction it does is the existence of these supersymmetric partners (or effects that could be explained by interaction with supersymmetric partners). These haven't been found. When supersymmetry appeared, they predicted that the lightest supersymmetric particle was just out of range for the best colliders at that moment. They have stayed with this prediction, the lightest supersymmetic particle is supposed to be just out of range of our
current colliders.

Some (Many?) believe that the supersymmetric community has been busy moving the goalposts, to keep supersymmetry on the horizon, instead of admitting that it just hasn't worked out.