I take some pity on the passenger, though it sounds like she may have handled it poorly and gone too confrontational throughout the process. Gate agents allow so many bags to skirt the limits. (I'm willing to wager my rollaboard wouldn't truly fit within the actual contraption).
United encourages high value, fragile items to be carried on (along with guitars, violins, etc ) and specifically mentions tennis racquets:
High-value, fragile and perishable items
United recommends that you do not pack high-value, fragile or perishable items in your checked baggage. United will accept such items as carry-on baggage (subject to carry-on baggage allowances) or as checked baggage (subject to checked baggage allowances). If you choose to pack high-value, fragile or perishable items in or as checked baggage in connection with travel within the United States, United is not liable for the loss of, damage to or delay in delivery of such items.
Examples of high-value, fragile or perishable items for which United is not liable (in the case of travel within the United States) or for which United's liability may be limited (in the case of most international travel) include, but are not limited to:
- Totally unprotected items such as tennis racquets and umbrellas, either individually checked or tied/strapped to the outside of luggage