The term 'Big Name in Cups' refers to three popular brands of disposable cups: Dixie (5 letters), Reeses (6 letters), and Reese (5 letters). Dixie is a brand of paper plates, cups, and bowls that has been around since the 1930s. Reeses is a brand of disposable cups and lids that are made from recycled materials. Reese is a brand of plastic cups that are designed to be durable and leak-proof. All three brands are well-known for their quality and reliability.