Chronic iCloud iMessage Storage full error!

tjpsnj,

Your messages can't be managed from the iCloud website. That has to be done on one of your devices that have the messages on them. It also sounds like your iCloud storage is full rather than your internal storage. That would mean that either your iCloud backups or content that you have in iCloud can't be backed up until more space is purchased or created. If you check your storage and notice that Messages are taking up a lot of space, here's information from the Manage your iCloud storage resource that will show you how to delete attachments from threads you don't need stored anymore:

"Delete texts and attachments in Messages

When you use Messages in iCloud, all of your text messages and the attachments you send and receive use your iCloud storage. You can make more space available when you delete texts and attachments—like photos, videos, and files—that you don't need anymore. 

Delete a message or attachment on your iPhone or iPad

1. In a message conversation, touch and hold the message bubble or attachment that you want to delete.
2. Tap More.
3. Tap Delete , then tap Delete Message. If you want to delete all messages in the thread tap Delete All, in the upper-left corner. Then tap Delete Conversation.

Delete multiple attachments from a contact or conversation

1. In a message conversation, tap the contact's name at the top of the screen.
2. Tap Info .
3. In the Photos section, tap See All. 
4. Tap Select, tap the images you'd like to delete.
5. Tap Delete, then tap Delete Attachment to confirm. 

Delete an entire conversation 

1. Swipe left over the conversation that you want to delete.
2. Tap Delete.
3. Tap Delete again to confirm. 

Delete a message or attachment on your Mac

1. Open a message conversation.
2. Control-click the blank area of a message bubble that you want to delete.
3. Choose Delete.
4. Click Delete to confirm.

Delete multiple attachments from a contact or conversation

1. Open a message conversation.
2. Click the Info button .
3. Scroll to the Photos section.
4. Click on an attachment then hold the Command key to select multiple attachments. 
5. Control-click or right-click, then choose Delete.
6. Click Delete Attachment to confirm. 

Delete an entire conversation

1. Control-click a conversation.
2. Choose Delete or Delete Conversation.
3. Click Delete to confirm."

You may also find some helpful information here: What's the difference between device storage and iCloud storage?

If things persist you can reach out for additional support here: Get Support

Take care.

Hey tjpsnj,

To manage your iCloud storage you're able to do this from your device and some from the iCloud website. Take a look at the following guide for steps on managing your iCloud storage and where you can do it: Manage your iCloud storage - Apple Support

With question 2, could you provide some. additional information on what you're looking for as it's a bit unclear.

Thanks for participating in Apple Support Communities.

Take care!