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They technically can, but it very fact specific for it to actually happen and a lot depends on how long ago the discharge took place.
There are two boards that consider discharge upgrades. The DD Form 293 discharge upgrade board (which can upgrade for up to 15 years) and the DD Form 149 records corrections board (which can hear a request longer than 15 years after, but which can also refuse to hear it based on the 3-year rule...meaning they usually take cases for only up to three years after you became aware of the issue with the record).
Both boards though apply the same standard. You have to prove that the original discharge status was either an error or it was an injustice at the time it was given. As you can imagine, very few situations fit that standard and they are very stingy with any such upgrade.