Discover Claim
I made a claim with discover for a charger which stopped working.

I had to call them, then was sent a claim form to do online. They wanted a receipt of original purchase, the warranty documentation, a form signed in ink then uploaded, a repair estimate, a copy of my discover statement the purchase was made on. I called Apple and was told they don't repair or replace out of warranty chargers, I would have to buy a new one. I called the warranty number and told them about this and the person said no problem, just make a note of that when you upload the signed form (which I did).

I uploaded all the documents, and was requested to provide the full discover statement rather than just the page with the transaction on and I complied.

Then I get a request a month later saying they want my repair estimate. I then call the adjustor 3-4 times and they never pick up and say leave a message and they will call back on the same day, so I left one every time for 2 weeks and never received a call back. I then called Discover and they called the warranty department and was told my adjustor wasn't in and they would request her to call back. Never received the call.

Fast forward to another request for the repair estimate, and I find out they had changed my adjustor so I call and I finally get ahold of my adjustor who then told me that I was not calling her and she was always my adjustor (which on prior emails was not the case). She then says I have to get a repair estimate and the only place is in store at an apple store nearly 2 hours away from me. I even asked if they had a mail in center which could verify and give me a repair estimate, and she said only Apple in store. From my calls with Apple it sounds like there wasn't a point in going in store anyways, as they won't repair. She said without me driving there and getting it they can't process the claim. She was rude and when I said they had never called me she told me just because I didn't want to drive I shouldn't accuse them of not returning my calls.

Now I've spent about 4-5 hours on the claim and gave up. I had probably also spent hours complaining about them. This was the same time I had problems with Discover bank, who constantly have their checks arriving late to the payee so I had to stop using them.

Amex Claim
Now my phone died late last year and I didn't have time to make a warranty claim with Amex (I used my Amex EDP), and I finally got round to doing it. A week after submitting a claim online, they requested a receipt of purchase, then one week later I got a statement credit for 400ish dollars.

So all in all, Amex is way better with claims than Discover. So much I don't care about the rewards. Not sure if this helped but I've been an Amex Platinum member in the USA for 2 years and prior to that in the UK since 2009.