Can same runner be put out twice?

[John Robertson]

This nearly happened in a game tonight. Here's the situation:

Two out. Runners at first and third. Batter connects for an extra-base hit but misses first base. Both runners ahead of him score, but he is thrown out at third base trying to stretch his hit into a triple. This is the third out of the inning. I know the defensive team is permitted to make an appeal for a fourth out to nullify a run, but can they do it on a runner who has already been put out at another base? My opinion would be to allow the appeal in order to nullify the two runs that scored.