This is not true. As noted upthread, musical instruments should be allowed in the overhead on a BE fare per published UA and DOT policy, assuming there is no other item carried on board and there is room in the overhead during boarding. See FAQ here.

Here is a Washington Post article with further information from pax. Article also (unlike upthread) says it was at the gate, so we clearly do not have correct information yet. Pax reports here that she immediately asked "what are my options?":

Once again, at the check-in counter this violates written UA policy (presumably her suitcase was checked, either at the counter or at the gate, with pax paying appropriate fees). At the gate, there may have been no room in the overhead and thus "no options." At that point, I would like to think the gate agent would then have a discussion with cabin crew about the possibility of stowing the instrument safely, to avoid alienating a customer who will abandon a nonrefundable ticket. Of course, this is not required of UA.

Washington Post reports pax booked onto AA instead to arrive to her destination on time with violin in cabin, no doubt eating the BE fare and the last-minute ticket.

FTers tend to have pretty strong priors that folks who get into trouble on airlines just don't know the rules, and that's fine, I suppose, as far as that goes. But this musician (or any musician) carrying an instrument is not just trying to get away with something and clearly does not lack willingness to pay to avoid a situation where she is parted from her instrument in flight.