  Although supernova remnants remain the main suspects as sources of Galactic
cosmic rays up to the knee, the supernova paradigm still has many loose ends.
The weakest point in this construction is the possibility that individual
supernova remnants can accelerate particles to the rigidity of the knee, $\sim
10^{6}$ GV. This scenario heavily relies upon the possibility to excite current
driven non-resonant hybrid modes while the shock is still at the beginning of
the Sedov phase. These modes can enhance the rate of particle scattering
thereby leading to potentially very-high maximum energies. Here we calculate
the spectrum of particles released into the interstellar medium from the
remnants of different types of supernovae. We find that only the remnants of
very powerful, rare core-collapse supernova explosions can accelerate light
elements such as hydrogen and helium nuclei, to the knee rigidity, and that the
local spectrum of cosmic rays directly constrains the rate of such events, if
they are also source of PeV cosmic rays. On the other hand, for remnants of
typical core-collapse supernova explosions, the Sedov phase is reached at late
times, when the maximum energy is too low and the spectrum at very-high
energies is very steep, being mostly produced during the ejecta dominated
phase. For typical thermonuclear explosions, resulting in type Ia supernovae,
we confirm previous findings that these objects can only produce cosmic rays up
to $\lesssim 10^{5}$ GeV. The implications for the overall cosmic ray spectrum
observed at the Earth and for the detection of PeVatrons by future gamma-ray
observatories are discussed.
