If you use CRM Online, and you run out of space, what happens?

This:

The Quick Create links disappear, replaced by a gray bar.

And the “New” button disappears from views and grids.

When this happens, either add space or delete records (like completed system jobs) to free up memory. After doing that, there will still be some delay before you can once again add records. If it continues to not allow record creation, open a support case with Microsoft support.