The short answer is, when it is a harmless error, meaning one where the result would have been the same even if the error had not been committed. Innumerable harmless errors have saved trial-court decisions from reversal on appeal over the years.
What is harmless is of course more complicated. The question recently came to a head when a judge granted a new trial in a car-accident case, even though he shouldn’t have. But applying an erroneous legal standard does not, alone, lead to an automatic reversal. This is because the judge and jury below are still owed deference in determining credibility (who lied or told the truth) and what the ultimate facts were (what actually happened), regardless of legal errors.
This imposes on appellate courts the duty of an oracle: would the trial court still have granted a new trial if it had applied the correct legal standards? If so, its decision must stand, and the appeal will be denied. In old English Law, this was known as the rule of the drunken coachman, evoking the image of the driver getting to the right destination by taking the wrong road. In modern American Law, it remains one more consideration the skilled appellate attorney must take into consideration.