How on earth can I increase this limit?

You’ve reached the current usage cap for GPT-4. You can continue with the default model now, or try again after 6:11 PM. [Learn more]

My current usage level is 4.
If I promote this to level 5, will the limitation go away?

There are two different types of usage limits:
One for ChatGPT and one for the API.
From what you are describing you are encountering issues with the limit for ChatGPT but are expecting that your usage tier for the API has any effect on this, which it doesn’t.

As of today you would need to upgrade to a Team account to get 100 messages every three hours.

Also, you might be doing your own math to come up with what you think is your “usage level” - however ChatGPT Plus subscription is not a consideration in API billing; past API payments for the use of API models is.

You can see your API account and tier level at the top and bottom of this page.

If you do develop an API application where you pay per-use, your limits for GPT-4 can be more like 40 requests per second instead of per three-hour.