Thermoelastic instabilities (TEI) occur in sliding bodies at sufficiently high speed because a small thermoelastic disturbance tends to localize the contact, leading to “hot spots.” The role that wear plays in TEI has been studied briefly and only on highly idealized cases. We extend and complete in detail a model of Dow and Burton who studied the specific configuration of a blade sliding on a rigid half-space normal to its line of contact. We find there is a limit value of wear coefficient that can be estimated by simple equations, above which TEI is completely eliminated. In the limiting case of non-conducting half-space, it depends linearly on thermal expansion, diffusivity, and friction coefficient and inversely on the conductance of the material of the sliding body. This may not always be in the practical range, but when considering conductance of the half-space, the limit wear can be lowered arbitrarily so as to be viable. In some applications, it may be possible to increase wear to reduce or suppress TEI. Hence, the common assumption of neglecting wear in simulations of sliding contacts with TEI and hotspots should be taken with care, and the present results give some important benchmarks.