You absolutely could, but most of the edges of the paved roads have completely vertical drops on the low sides. I see a few smaller dirt bikes around like Honda XR's and they also have a variety of Enfield's with knobbies and spoked rims.
Stay off the roads and there is unlimited off-road riding, kind of like riding in the southwest states but at high elevation.
Nepal (which is one of the most mountainous countries in the world), has tons of bike rental places but it seems much less common in Tibet because you must have a guide outside Lhasa. Our guide in Tibet leads multi day rides on BMW GS's for small groups and up to 20 riders!