Yes it is done regularly on recurrent sim training, but not to full travel because it is normally treated as a control system fault that can be stopped before it becomes totally uncontrollable, and the Alaska Airlines incident was a mechanical failure of the acme threads on the screw jack where the stab was free to tilt up as far as it could go and they were doomed.
In sim training usually the instructor will generate a stab runaway while you're in some settled down situation minding your own business, and you have to recognize it (there is usually a clacker sound that goes off after it has run for more than say 5 seconds if you don't notice the indication moving), hit the stab trim DISC button on the wheel to disengage the trim channels, and do what ya gotta do. If the autopilot was on in ALT hold mode it would have disengaged itself at some point also when the elevator force required to hold altitude got too high and exceeded the torque limit on the autopilot servo.
What you are now faced with is an airplane that is now trimmed for a single speed, wherever the stab stopped at, and to fly at any other speed requires you to displace the elevator and hold it there, against the springs in the pitch feel system wanting to return it to neutral. So it all depends on how far the stab moved before you cut it off.
Nose down runaways are generally worse than nose up runaways, because it's easier to push and hold then pull and hold, because you can jam your leg against the column to give your arms a rest on a nose up runaway. Stab runaways are one of the more unpleasant training scenarios especially because if you get a nose down runaway (which depends on the evilness quotient of your instructor) your arms are going to be really sore when you're done.
Most airplanes have significant trim change with flap extension and flaps can be deployed to help with controlability depending on the airplane's response to flap deployement, so that's a case where knowing your airplane can really help.
Although the AA261 incident did absolve the jackscrew grease as a root cause, it did cause a review and testing of grease intermix practices (the jackscrew had both lithium grease and clay base grease on it and the grease was degraded), where it was found that mixing lithium base and clay base greases caused premature breakdown, and an industry wide campaign was done. Most maintenance manuals have admonishments against mixing clay and lithium grease.
Problem is, there isn't a separate mil-spec for clay and lithium, only between moly and non-moly grease, and greases may only be called for by the mil-spec not brand name, so the situation, to this day, can be bit of a mess. For example, Aeroshell 33 (lithium) and Aeroshell 7 (clay) are the same mil-spec, but are not supposed to be intermixed. Clear as mud?