No, free checked bags is not a benefit of The Platinum Card® from American Express. However, the Amex Platinum has an annual $200 airline fee credit to cover incidentals like checked bag fees for your designated airline.
This credit can be used for other incidentals like seat assignments, lounge passes, change fees, pet travel, and in-flight purchases.
If you fly Delta Air Lines, you may want to consider getting the Delta SkyMiles® Gold American Express Card or Delta SkyMiles® Platinum American Express Card to get free checked bags on your next trip.