A study of the thermal conductivity of single crystal specimens of pure tin and indium has been made in the temperature range 2 to 4.2 $^\circ$K in both the normal and superconducting states. Values of the normal state thermal conductivity, corrected for magnetoresistance, fitted well the expression 1/K=$\alpha$T$^2$+$\beta$/T, although deviations from this formula were observed in the purest specimens. Serious departures from Matthiessen's rule occurred, however, in that the magnitude of the lattice resistance ($\alpha$T$^2$) depended strongly on purity. Systematic variations in the ratio of conductivities K$_s$/K$_n$ with purity were found to follow the simple expression suggested by Hulm. The limiting curves for K$_s$/K$_n$ (in the cases of all impurity scattering, and of all lattice scattering of electrons) are compared with recent calculations on the Bardeen-Cooper-Schrieffer theory of superconductivity.