I don't understand how that can be the case. Let's say you work as a plumber. The median income is 56k, meaning half the workforce earns less than 56k. There are two ways this can be the case: (1) the field is rapidly expanding and the median is thus artificially low by all the new entrants; (2) the salary of a mid-career plumber is 56k.
You cannot have high salaries over the board (save for the few percent new entrants who have not yet paid their dues) and 56k median income at the same time. Either the public perception about trade salaries is wrong or the BLS is wrong. I don't know which one is the case, but in the absence of better heuristics, I trust BLS more than anecdotes.