Shear thickening is studied in suspensions consisting of micrometer sized polystyrene spheres dispersed in Boger fluids. In comparison with available data for suspensions in Newtonian media, shear thickening occurs at much lower volume fractions. Suspensions in normal nonlinear viscoelastic media typically do not display shear thickening at all. In suspensions in Boger fluids, the onset of shear thickening is shown to be governed by a critical shear stress. The effect of particle size is consistent with a scaling with the particles radius to the third power, as for Brownian hard spheres. The critical stresses are, however, orders of magnitude higher than for similar suspensions in low viscosity, Newtonian media. The first normal stress coefficient is also affected by the presence of particles, but it displays shear thickening as well as shear thinning. No specific microstructural features have been observed during flow, neither alignment nor hydroclustering seem to occur. Shear thickening in Boger fluids is possibly caused by the enhanced hydrodynamic interparticle interactions, related to the specific viscoelastic properties of the suspending medium, i.e., the absence of shear thinning and pronounced extensional hardening. This seems to be confirmed by recent simulation results.