Yup. If the cooler is in one tank of the radiator, the tube could be rusted through. Cooling system pressure is 15 pounds when it's hot and that is typically higher than the pressure in the transmission cooler lines. Even if the transmission pressure is higher, it will be 0 pounds with the engine shut off while the cooling system will stay pressurized until it cools down in a few minutes.
You will also usually find a brownish mud inside the cooling system reservoir. Transmission fluid will go into the coolant before the cooling system pressure builds up. That mud will rot radiator and heater hoses.