Hi Steve, Andy,
Provided you have good lines to work with it is fairly straightforward to measure radial velocities to a precision 1/10 of the resolution or better. (As I plan to show at the workshop in a couple of weeks) so an ALPY with say 10A resolution could measure a 1A shift or 50km/s at 6000A as a reasonable target.
You can go much, much more precise than this though if you use an off telescope fibre fed spectrograph and correlation techniques as is done for exoplanets. eg Buil measured to 50m/s (yes metres!) using a 0.6A resolution echelle spectrograph which I think works out at ~1/500 of the resolution