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It depends.
Some issuers ask for proof, some trust your word.
What constitutes a "student"? Attending a four year university? Grad school? Community college? Culinary Institute? What if you are only taking a class to keep up to date on design and drafting software?
A student is anyone enrolled in a college or university that can prove it with a valid registration. That includes community college. Read the fine print on any application to see what youn need to qualify. I don't think a simple extension class would qualify. But taking graduate courses at night after work to get a masters would qualify since your registration would be the same as a full time grad student who has never worked.
Be aware that student applications are typically for people with no or short credit histories, so if you have a long history, they may look twice at it.
Wells Fargo didn't. Stayed $750 CL til I closed it. My student Discover eventually had a $12.2k CL and became a Platnium Discover (now would be a More Card) before I lost it in 2003. I had 2 other VISAs from banks that grew nicey with me until I closed them years latter (one had $2500CL and the other $4500).