Up to around 5 years ago, even economy passengers were allowed two checked in bags, each up to 32kg in weight on most carriers between the UK & USA (both ways). Now most carriers limit economy passengers to one 23kg bag.
I can only assume that in the past with so many transatlantic flights, no one worried about the amount being carried in the hold. But there is money to be made out of those carrying excess baggage and less fuel consumed by a lighter aircraft.
Specifically answering your question – No, you are limited to one 23kg bag (or whatever).
One crumb of comfort I can offer is that check-in staff have told me that the 23kg limit is actually 25kg (to allow for errors in your bathroom scales).
In general, as you go up the seating/passenger grades, your luggage allowance is increased – check the terms & conditions of your carrier.