I really don't think it can. Looking through examples of people using that construction [1] they are all being used to emphasize the speaker's lack of something, without any implication that if someone were to have one it would be the speaker. Do you see any examples of someone using it your way?
> Good luck with your ESL tests as well.
I don't know if it's apparent to you how condescending that sounds? But, for what it's worth, I'm a native speaker of English and have a degree in linguistics.
[1] https://www.google.com/search?q=%22i+don%27t+even+have%22