I bought a nonrefundable ticket and couldn't fly because of a panic attack that happened at the airport. Can I still get credit/money back?

I was told they would charge me $200 to change each ticket--which is about how much the tickets cost me. Which is preposterous. Other airlines charge $20-$40 to do this in many cases.

Added (1). American Airlines

Cost to change a ticket is based on fare class and $200 is on the high side of typical, but $75 each way is normal. What airline charges $20 to $40 to change a non-refundable fare?
Nonrfundable means no refunds, and a panic attack is not their fault.
Next time pay their insurance cost at purchase or buy a much more expensive fare class.
American, United, and Delta are all at $200.
Allegiant is $75 each way, and issues credit only towards another ticket
Spirit gets at least $90 and has other fees
Southwest airlines features included baggage and no change fees.
Jetblue at least $75 and up to $150 depending upon initial fare.
United Airlines introduced Basic Economy, non-refundable tickets. They show the fees in the terms and fare class, and is generally $200 also now.
As to your disability, argue it out with them. In court, you would have to prove your disability affects your understanding of "nonrefundable".

This is all in contract law and airlines make their money on fees now. The change fee amounts are in the fare class rules, and are not a specific number any more.
United is now $200 on their nonrefundable tickets called Basic Economy, started at the end of 2016

No you will get nothing back. Take the Greyhound bus next time.

YOU have to read the Terms and Conditions that YOU agreed to when YOU bought the tickets. I did not buy them so I have not read them.

So you snoozed through English class. You get to pay for that.

What part of NONREFUNDABLE don't you understand?

You must be living in a timewarp if you think "other" airlines are charging $40 or less for change fees. Your idea of "recent years" and most people's are not the same thing. This is a standard practice since the early 2000s.
You purchased a non-refundable ticket, which means you pay the charge fee or you lose the ticket.

If you have a condition that is governed by ADA, you need to learn how to accommodation regulations work. The ADA is not a "get out of jail free" card. If you want an accommodation, you need to request it BEFORE you need it.
- Prior to purchasing the ticket you would have needed to request the accommodation. My guess is that the airlines are not required to accommodate the ADA in this manner since there are different rules for individuals with medical issues impeding the individual's ability to fly.

Nonrefundable ticket. Did you fail to understand nonrefundable ticket part?

A change fee of $200 is pretty typical. Many airlines will offer same-day changes for $75, but only if you are flying at a different time that same day. You say you've changed tickets for $40, but perhaps it was a different fare class; I have to say I've never heard of such a cheap change fee, so your experience in the past is not typical. And it doesn't change the fact that you agreed to these terms when you booked. When you book a nonrefundable ticket, then you are bound by the terms of that ticket - this is always disclosed in the terms and conditions that no one bothers to read. In this case, it sounds like it would have been best to just cancel and not seek out a change. The ADA doesn't mean that these change fees don't apply to you. If you have a disability that might affect your ability to travel at the last minute then you should purchase travel insurance that will cover the cost of your fare in the case of such an incident.

No, you can't if it's non refundable

I LOVE to see what airline charges $20-$40 change fee - Southwest is the only one, otherwise you are delusional