Please check that you do not have any filters processing your incoming email in Thunderbird. A filter would explain why the same emails move back to trash after you drag them out. Also check online in your webmail that those messages are not put in the trash by the online email before they are even downloaded to Thunderbird. Please report what you find.

It wasn't happening until a few days ago and I didn't change anything to any filters that might process the incoming email in Thunderbird. I also didn't change anything online with my gmail accounts which is where it is happening. It also isn't just that I can't move some emails from trash back to the inbox. It is that something is moving emails from my inbox into trash before I even see them. What is even weirder is that it doesn't happen with all my emails. In fact, the ones that seem to be moving are from companies like Amazon, Newsy, The New York Times, and Enterprise.

That sounds like something, maybe in Gmail, is deciding those are spam. I understood you can move them from trash but they just go back there again, which sounds like a filter acting on the folder you put them back into. However it may also be automatic spam fitering in either Gmail or Thunderbird. If you use IMAP and Gmail moves them to trash, Thunderbird syncs with Gmail and moves them too, or if Thunderbird moves them, Gmail syncs and puts them in trash. To distinguish which one is doing the moving - leave TB closed, just use Gmail for a while until a few of these kinds of message have arrived as new mail. See where they are in Gmail before opening TB. If they are already placed in Trash, Gmail is doing it. If they were in Gmail Inbox, you then open TB and it downloads, then they arrive in Trash, it's something in Thunderbird. Tricky because you may not know from looking at new mail whether it is likely to be affected or not. Filters exist in both, automatic spam detection exists in both, and Thunderbird also has account settings that might be set to place some incoming messages directly in Trash.

Okay, I will try what you said. Even stranger news for you, however, I have not tried to move anything from trash back to inbox since I posted my question but I just looked at my inbox and the files I tried to move from trash to inbox that went right back to trash are now in my inbox. Very, very strange at least to me. I will post after I receive some emails that were moving directly to trash once I see what happens while I access my accounts online instead of using Thunderbird. Thanks for all your help.