Not sure if OP is asking about "checking cabin bags" at the baggage check/ticketing desk, or at the gate. Certainly at the baggage check they will stick to the ticketed baggage allowance, regardless of the size of the bags. But at the gate, at least for US full service carriers, they are almost always willing to gate-check a bag for free because it results in more overhead space for others and faster boarding (yes, more of a thing on narrowbodies). Many times they encourage pax to check bags at the gate and certainly the last handful of pax boarding might be required to gate-check.

So I don't think anyone has unequivocally answered here whether BA at JFK would gate-check a bag for free that isn't part of the baggage allowance.