Hi Daniel.Escudero,
Thanks for your post and let us know above information.
From your description, please sign out Teams desktop client and exit Teams completely. Go to %appdata%\Microsoft\teams\Cache and delete all the files. Then go to %appdata%\Microsoft\teams\Application Cache\Cache, delete all the files. Re-open Teams to see
if this will make any improvement.
Note: please don't forget to copy all the files to a different location before you delete them.
Also, to narrow down this issue, you may try to
perform
a clean boot in Windows and check if there is the seem results.
In addition, as you mentioned this issue happens when several messages are incoming from different chats, to help you further, we would like to confirm the following information with you for analysis:
1. Please sign into Teams web APP to see if there is the same issue when receiving several messages.
2. Can you receive the chat messages?
3. Is there any error messages if you click to check the chat message? If so, please send us a screenshoot of the error message.
Best regards,
Jennifer