Most galactic cosmic rays are probably accelerated in the blast waves of supernova remnants. This doesn't mean that the supernova explosion itself gets the particles up to these speeds. The remnants of the explosions, expanding clouds of gas and magnetic field, can last for thousands of years, and this is where cosmic rays are accelerated. Bouncing back and forth in the magnetic field of the remnant randomly let's some of the particles gain energy, and become cosmic rays. Eventually they build up enough speed that the remnant can no longer contain them, and they escape into the Galaxy.
Because the cosmic rays eventually escape the supernova remnant, they can only be accelerated up to a certain maximum energy, which depends upon the size of the acceleration region and the magnetic field strength.
However, cosmic rays have been observed at much higher energies than supernova remnants can generate, and where these ultra-high-energies come from is a big question. Perhaps they come from outside the Galaxy, from active galactic nuclei, quasars or gamma ray bursts. Or perhaps they're the signature of some exotic new physics: superstrings, exotic dark matter, strongly-interacting neutrinos, or topological defects in the very structure of the universe. Questions like these tie cosmic-ray astrophysics to basic particle physics and the fundamental nature of the universe.