jbrehm,
Depends on the size of the turbo, the size of the bearings and shaft, and type of bearings (full floating hydrodynamic or rolling element).
A turbo for a (relatively) large CAT 3612 may have greater axial and radial play specs that a turbo for an 800cc smart car diesel.
For typical hydrodynamic journal and thrust bearing systems, and assuming that you are thinking typical pass car engine size, I would guess 0.001 to 0.003 axial play is OK (units are inches), and 0.001 to 0.004 radial play is OK. True radial play is difficult to measure without the correct probe. Accurate radial play should be measured as shaft excursion between the bearings when manually displacing both the compressor and turbine ends of the rotating group in the same direction.
Dick