Is it possible to get my exchanged iPhone back?

As stated by one poster who was correct, no you can’t get your traded in device back.

It states that on the trade in sight.

once you ship it to them that’s it.

The only exception is if they lower the value and you reject the trade in.

Apple Trade In - Apple

Can I cancel a trade-in?

If you haven’t dropped off or shipped your device yet, you can cancel your trade-in by simply keeping your device.

If you’ve already sent us your device, the trade-in can’t be cancelled. However, if you receive a lower revised trade-in value after we’ve inspected your device, you can choose to reject it.

It sounds like you took advantage of a trade-in program. I would guess that it is too late to get your old phone back. I say "guess" because this is a user-to-user forum – you are not speaking to Apple or any retailer here.

In terms of buying another phone of a similar kind to the old one, you could look in the Apple Certified Refurbished Store. At the moment, they seem to have 14-series models and to be out of 13-series ones.

Refurbished iPhone Deals - Apple

To make things perfectly clear, my reference to the Certified Refurbished Store was not a claim that the OP could cancel the trade-in deal, and get the old phone (or one exactly like it) back – no harm, no foul, no pocketbook hit. The reference to the Certified Refurbished Store was simply to note that older-model phones may still be for sale there. As in, it might be possible to find a new instance of the old model at additional expense.

The OP is probably better off learning to live with whatever iPhone 16 series model they purchased. Especially if their beef is not with the iPhone 16 series hardware per se, but with iOS 18. Apple says that Certified Refurbished iPhones come with "the original operating system or a more recent version". A Certified Refurbished phone could come with iOS 18 and it would then be impossible to downgrade the iOS version.

No.

Edit....

"Is it possible to get my exchanged iPhone back?: Is it possible to get my old phone back. 13 pro max plus. I am not happy with the 16 at all."

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By "16", I take it you are referring to "iPhone 16". Is that correct?

iPhone 16:

So, what is it that you don't like about your iPhone 16? If having issues, reply back with what they are, to see if things can be fixed.