Thermal conductivity and specific heat were measured in the superconducting state of the heavy fermion material Ce_{1-x}La_{x}CoIn_{5}. With increasing impurity concentration x, the suppression of T_{c} is accompanied by the increase in the residual electronic specific heat expected of a d-wave superconductor, but it occurs in parallel with a decrease in residual electronic thermal conductivity. This contrasting behavior reveals the presence of uncondensed electrons coexisting with nodal quasiparticles. An extreme multiband scenario is proposed, with a d-wave superconducting gap on the heavy-electron sheets of the Fermi surface and a negligible gap on the light, three-dimensional pockets.