  The rapid rise in the cosmic ray positron fraction above 10 GeV, as measured
by PAMELA and AMS, suggests the existence of nearby primary sources of high
energy positrons, such as pulsars or annihilating/decaying dark matter. In
contrast, the spectrum of secondary positrons produced through the collisions
of cosmic rays in the interstellar medium is predicted to fall rapidly with
energy, and thus is unable to account for the observed rise. It has been
proposed, however, that secondary positrons could be produced and then
accelerated in nearby supernova remnants, potentially explaining the observed
rise, without the need of primary positron sources. Yet, if secondary positrons
are accelerated in such shocks, other secondary cosmic ray species (such as
boron nuclei, and antiprotons) will also be accelerated, leading to rises in
the boron-to-carbon and antiproton-to-proton ratios. The measurements of the
boron-to-carbon ratio by the PAMELA and AMS collaborations, however, show no
sign of such a rise. With this new data in hand, we revisit the secondary
acceleration scenario for the rising positron fraction. Assuming that the same
supernova remnants accelerate both light nuclei (protons, helium) and heavier
cosmic ray species, we find that no more than ~25% of the observed rise in the
positron fraction can result from this mechanism (at the 95% confidence level)
