You can apply for an extension of stay or a change of status at any time before your I-94 expires.
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The request must be "nonfrivolous," that is, it must have some basis for possibly being approved. If you timely file this request (that is, you file before you current status expires), then you are in what is known as an "authorized period of stay." You are not in your previous status, but you are not out of status, and that's the important thing.
I suspect the reason for saying 45 days prior is so that people don't forget, or don't run up against the expiration deadline.
This post is not legal advice, and does not create a lawyer-client relationship.
Yes, if you have an ordinary visa I-94 (weren't admitted on Visa Waiver), you can apply to extend your stay, but the application must be received by USCIS before the I-94 expires, not just mailed before.
There is no requirement that the application must be made 45 days in advance of the expiration. It will take about three months to be adjudicated. If you have any evidence of your health situation (diagnosis or letter from a doctor), definitely include that - but don't delay sending the application in to get this; better to send it later in response to an inquiry.
Speak with a lawyer directly for further guidance.